Revisiting example 106 from Steen and Seebach

Posted on December 25, 2021 by Dan Ma

The example 106 from Counterexamples in Topology by Steen and Seebach [3] is the space $\omega_1 \times I^I$ where the first factor $\omega_1$ is the space of countable ordinals with the usual order topology and the second factor $I^I$ is the product of continuum many copies of the unit interval $I=[0,1]$ .

This space was previously discussed in this site. One of the key results from that discussion is that $\omega_1 \times I^I$ is not normal, a result not shown in Steen and Seebach. The proof that was given in this site (see here) is based on an article published in 1976 [1], long before the publication date of the first edition of Steen and Seebach in 1970. It turns out that the non-normality of $\omega_1 \times I^I$ was given as an exercise in Steen and Seebach in the problem section at the end of the book (problem 127 in page 211, Dover edition). Problem 127: Show that $[0, \Omega) \times I^I$ is not normal. This indicates that the result not shown in Steen and Seebach was because it was given as a problem and not because the tool for solving it was not yet available. The fact that it is given as an exercise also means that there is a more basic proof of the non-normality of $\omega_1 \times I^I$ . So, once this is realized, I set out to find a simpler proof or at least one that does not rely on the result from [1]. Interestingly, this proof brings out a broader discussion that is worthwhile and goes beyond the example at hand. The goal here is to examine the more basic proof and the broader discussion.

A Classic Example

Before talking about the promised proof, we consider the product of $\omega_1$ and its immediate successor.

As noted at the beginning, the space $\omega_1$ is the set of all countable ordinals with the order topology. The ordinal $\omega_1+1$ is the immediate successor of $\omega_1$ . It can be regarded as the result of adding one more point to $\omega_1$ . The extra point is $\omega_1$ , i.e., $\omega_1 +1=\omega_1 \cup \{ \omega_1 \}$ with $\omega_1$ greater than all points $\beta < \omega_1$ . The ordinal $\omega_1+1$ with the order topology is a compact space. Using interval notation, $\omega_1=[0, \omega_1)$ and $\omega_1+1=[0, \omega_1]$ . As ordinals, $\omega_1$ is the first uncountable ordinal and $\omega_1+1$ is the first uncountable successor ordinal. For more information, see here.

The product $[0, \omega_1) \times [0, \omega_1]$ is a classic example of a product of a normal space (the first factor) and a compact space (the second factor) that is not normal. This example and others like it show that normality is easily broken upon taking product even if one of the factors is as nice as a compact space. The non-normality of $[0, \omega_1) \times [0, \omega_1]$ is discussed here. In that proof, two disjoint closed sets $H$ and $K$ are given such that they cannot be separated by disjoint open sets. The $H$ and $K$ are:

$H=\{ (\alpha, \alpha): \alpha < \omega_1 \}$

$K=\{(\alpha, \omega_1): \alpha < \omega_1 \}$

The Basic Proof

To show that $\omega_1 \times I^I$ is not normal, we show that one of its closed subspaces is not normal. That closed subspace is $[0, \omega_1) \times [0, \omega_1]$ . To this end, we show that $[0, \omega_1]$ can be embedded in the product space $I^I$ . With a non-normal closed subspace, it follows that $\omega_1 \times I^I$ is not normal. The remainder of the proof is to give the embedding.

We show that $[0, \omega_1]$ can be embedded as a closed subspace of $I^{\omega_1}$ , the product of $\omega_1$ many copies of $I$ . This means that $[0, \omega_1]$ is also a closed subspace of $I^I$ .

For each $\beta < \omega_1$ , define $T_\beta: \omega_1 \rightarrow I$ as follows:

$T_\beta(\gamma) = \begin{cases} 1 & \ \ \ \mbox{if } \gamma < \beta \\ 0 & \ \ \ \mbox{if } \gamma \ge \beta \end{cases}$

Furthermore, define $T: \omega_1 \rightarrow I$ by letting $T(\beta)=1$ for all $\beta < \omega_1$ . Consider the correspondence $\beta \rightarrow T_\beta$ with $\beta < \omega_1$ and $\omega_1 \rightarrow T$ . The mapping is clearly one-to-one from $[0, \omega_1]$ onto $\{ T_\beta: \beta < \omega_1 \} \cup \{ T \}$ . Upon closer inspection, the mapping in each direction is continuous (this is a good exercise to walk through). Thus, the mapping is a homeomorphism. It follows that $[0, \omega_1]$ can be considered a subspace of $I^{\omega_1}$ . Since $[0, \omega_1]$ is compact, it must be a closed subspace. With the cardinality of $\omega_1$ being less than or equal to continuum, it follows that $[0, \omega_1]$ can be embedded as a closed subspace of $I^I$ .

Stone-Cech Compactification

The first broader discussion is that of Stone-Cech compactification. More specifically, $\beta \omega_1=\omega_1+1$ , i.e., the Stone-Cech compactification of the first uncountable ordinal is its immediate successor.

To see that $\beta \omega_1=\omega_1+1$ , note that every continuous function defined on $[0,\omega_1)$ is bounded and is eventually constant (see result B here). As a result, every continuous function defined on $[0,\omega_1)$ can be extended to a continuous function defined on $[0, \omega_1]$ . For any continuous function $f: \omega_1 \rightarrow \mathbb{R}$ , we can simply define $f(\omega_1)$ to be the eventual constant value. A subspace $W$ of a space $Y$ is $C^*$ -embedded in $Y$ if every bounded continuous real-valued function on $W$ can be extended to $Y$ . According to theorem 19.12 in [4], if $Y$ is a compactification of $X$ and if $X$ is $C^*$ -embedded in $Y$ , then $Y$ is the Stone-Cech compactification of $X$ . Thus $[0,\omega_1)$ is $C^*$ -embedded in $[0,\omega_1]$ and $[0,\omega_1]$ is the Stone-Cech compactification of $[0,\omega_1)$ . In this instance, the Stone-Cech compactification agrees with the one-point compactification. Consider the following class theorem about normality in product space. The theorem is Corollary 3.4 in the chapter on products of normal spaces in the handbook of set-theoretic topology [2].

Theorem 1
Let $X$ be a space. The following conditions are equivalent.

The space $X$ is paracompact.
The product space $X \times \beta X$ is normal.

Based on the discussion presented above, the non-normality of $\omega_1 \times I^I$ is due to the non-normality of $[0, \omega_1) \times [0, \omega_1]$ . Based on this theorem, the non-normality of $[0, \omega_1) \times [0, \omega_1]$ is due to the non-paracompactness of $[0, \omega_1)$ . See result G here for a proof that $[0, \omega_1)$ is not paracompact.

The discussion up to this point points to two ways to prove that $\omega_1 \times I^I$ is not normal. One way is the basic proof indicated above. The other way is to use Theorem 1, along with the homeomorphic embedding from $[0, \omega_1]$ into $I^I$ , the fact that $\beta \omega_1=\omega_1+1$ and the fact that $[0, \omega_1)$ is not paracompact. Both are valuable. The first way is basic and is a constructive proof. Because it is more hands-on, it is a better proof to learn from. The second way provides a broader perspective that is informative but requires quoting a couple of fairly deep results. Perhaps it is best used as a second proof for perspective.

Countable Tightness

The essence of the basic proof above goes like this: if the space $Y$ contains a copy of $\omega_1+1=[0, \omega_1]$ , then the product space $[0, \omega_1) \times Y$ is not normal. The contrapositive statement would be the following:

Corollary
Let $Y$ be a space. If the product space $\omega_1 \times Y$ is normal, then $Y$ cannot contain a copy of $\omega_1+1$ .

In the space of $\omega_1+1=[0, \omega_1]$ , note the following about the last point: $\omega_1 \in \overline{[0, \omega_1)}$ but $\omega_1 \notin \overline{C}$ for any countable $C \subset [0, \omega_1)$ , i.e., the last point is the limit point of the set of all the points preceding it but is not in the closure of any countable set. This means that the space $\omega_1+1=[0, \omega_1]$ does not have countable tightness (or is not countably tight). See here for definition. The property of countable tightness is hereditary. If $Y$ contains a copy of $\omega_1+1$ , then $Y$ is not countably tight (or is uncountably tight). This brings us to the following theorem.

Theorem 2
Let $Y$ be an infinite compact space. Then $\omega_1 \times Y$ is normal if and only if $Y$ has countable tightness.

Whenever we consider the normality of a product with the first factor being $\omega_1$ and the second factor being a compact space, the real story is the tightness of that compact space. If the tightness is countable, the product is normal. Otherwise, the product is not normal. The theorem is another reason that $\omega_1 \times I^I$ is not normal. Instead of embedding $[0, \omega_1]$ into $I^I$ , we can actually show that $I^I$ does not have countable tightness. This is the approach that was taken in this previous post.

Theorem 2 is the result from 1976 alluded to earlier [1]. A proof of Theorem 2 is found in this previous post. For results concerning normality in a product space with a compact factor (the other factor does not have to be $\omega_1$ ), see the chapter on products of normal spaces in the handbook of set-theoretic topology [2].

Reference

Nogura, T., Tightness of compact Hausdorff space and normality of product spaces, J. Math. Soc. Japan, 28, 360-362, 1976.
Przymusinski T. C., Products of Normal Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 781-826, 1984.
Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.
Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

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Lindelof Exercise 2

Posted on July 11, 2019 by Dan Ma

The preceding post is an exercise showing that the product of countably many $\sigma$ -compact spaces is a Lindelof space. The result is an example of a situation where the Lindelof property is countably productive if each factor is a “nice” Lindelof space. In this case, “nice” means $\sigma$ -compact. This post gives several exercises surrounding the notion of $\sigma$ -compactness.

Exercise 2.A

According to the preceding exercise, the product of countably many $\sigma$ -compact spaces is a Lindelof space. Give an example showing that the result cannot be extended to the product of uncountably many $\sigma$ -compact spaces. More specifically, give an example of a product of uncountably many $\sigma$ -compact spaces such that the product space is not Lindelof.

Exercise 2.B

Any $\sigma$ -compact space is Lindelof. Since $\mathbb{R}=\bigcup_{n=1}^\infty [-n,n]$ , the real line with the usual Euclidean topology is $\sigma$ -compact. This exercise is to find an example of “Lindelof does not imply $\sigma$ -compact.” Find one such example among the subspaces of the real line. Note that as a subspace of the real line, the example would be a separable metric space, hence would be a Lindelof space.

Exercise 2.C

This exercise is also to look for an example of a space that is Lindelof and not $\sigma$ -compact. The example sought is a non-metric one, preferably a space whose underlying set is the real line and whose topology is finer than the Euclidean topology.

Exercise 2.D

Show that the product of two Lindelof spaces is a Lindelof space whenever one of the factors is a $\sigma$ -compact space.

Exercise 2.E

Prove that the product of finitely many $\sigma$ -compact spaces is a $\sigma$ -compact space. Give an example of a space showing that the product of countably and infinitely many $\sigma$ -compact spaces does not have to be $\sigma$ -compact. For example, show that $\mathbb{R}^\omega$ , the product of countably many copies of the real line, is not $\sigma$ -compact.

Comments

The Lindelof property and $\sigma$ -compactness are basic topological notions. The above exercises are natural questions based on these two basic notions. One immediate purpose of these exercises is that they provide further interaction with the two basic notions. More importantly, working on these exercise give exposure to mathematics that is seemingly unrelated to the two basic notions. For example, finding $\sigma$ -compactness on subspaces of the real line and subspaces of compact spaces naturally uses a Baire category argument, which is a deep and rich topic that finds uses in multiple areas of mathematics. For this reason, these exercises present excellent learning opportunities not only in topology but also in other useful mathematical topics.

If preferred, the exercises can be attacked head on. The exercises are also intended to be a guided tour. Hints are also provided below. Two sets of hints are given – Hints (blue dividers) and Further Hints (maroon dividers). The proofs of certain key facts are also given (orange dividers). Concluding remarks are given at the end of the post.

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Hints for Exercise 2.A

Prove that the Lindelof property is hereditary with respect to closed subspaces. That is, if $X$ is a Lindelof space, then every closed subspace of $X$ is also Lindelof.

Prove that if $X$ is a Lindelof space, then every closed and discrete subset of $X$ is countable (every space that has this property is said to have countable extent).

Show that the product of uncountably many copies of the real line does not have countable extent. Specifically, focus on either one of the following two examples.

Show that the product space $\mathbb{R}^c$ has a closed and discrete subspace of cardinality continuum where $c$ is cardinality of continuum. Hence $\mathbb{R}^c$ is not Lindelof.
Show that the product space $\mathbb{R}^{\omega_1}$ has a closed and discrete subspace of cardinality $\omega_1$ where $\omega_1$ is the first uncountable ordinal. Hence $\mathbb{R}^{\omega_1}$ is not Lindelof.

Hints for Exercise 2.B

Let $\mathbb{P}$ be the set of all irrational numbers. Show that $\mathbb{P}$ as a subspace of the real line is not $\sigma$ -compact.

Hints for Exercise 2.C

Let $S$ be the real line with the topology generated by the half open and half closed intervals of the form $[a,b)=\{ x \in \mathbb{R}: a \le x < b \}$ . The real line with this topology is called the Sorgenfrey line. Show that $S$ is Lindelof and is not $\sigma$ -compact.

Hints for Exercise 2.D

It is helpful to first prove: the product of two Lindelof space is Lindelof if one of the factors is a compact space. The Tube lemma is helpful.

Tube Lemma
Let $X$ be a space. Let $Y$ be a compact space. Suppose that $U$ is an open subset of $X \times Y$ and suppose that $\{ x \} \times Y \subset U$ where $x \in X$ . Then there exists an open subset $V$ of $X$ such that $\{ x \} \times Y \subset V \times Y \subset U$ .

Hints for Exercise 2.E

Since the real line $\mathbb{R}$ is homeomorphic to the open interval $(0,1)$ , $\mathbb{R}^\omega$ is homeomorphic to $(0,1)^\omega$ . Show that $(0,1)^\omega$ is not $\sigma$ -compact.

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Further Hints for Exercise 2.A

The hints here focus on the example $\mathbb{R}^c$ .

Let $I=[0,1]$ . Let $\omega$ be the first infinite ordinal. For convenience, consider $\omega$ the set $\{ 0,1,2,3,\cdots \}$ , the set of all non-negative integers. Since $\omega^I$ is a closed subset of $\mathbb{R}^I$ , any closed and discrete subset of $\omega^I$ is a closed and discrete subset of $\mathbb{R}^I$ . The task at hand is to find a closed and discrete subset of $Y=\omega^I$ . To this end, we define $W=\{W_x: x \in I \}$ after setting up background information.

For each $t \in I$ , choose a sequence $O_{t,1},O_{t,2},O_{t,3},\cdots$ of open intervals (in the usual topology of $I$ ) such that

$\{ t \}=\bigcap_{j=1}^\infty O_{t,j}$ ,
$\overline{O_{t,j+1}} \subset O_{t,j}$ for each $j$ (the closure is in the usual topology of $I$ ).

Note. For each $t \in I-\{0,1 \}$ , the open intervals $O_{t,j}$ are of the form $(a,b)$ . For $t=0$ , the open intervals $O_{t,j}$ are of the form $[0,b)$ . For $t=1$ , the open intervals $O_{t,j}$ are of the form $(a,1]$ .

For each $t \in I$ , define the map $f_t: I \rightarrow \omega$ as follows:

$f_t(x) = \begin{cases} 0 & \ \ \ \mbox{if } x=t \\ 1 & \ \ \ \mbox{if } x \in I-O_{t,1} \\ 2 & \ \ \ \mbox{if } x \in I-O_{t,2} \text{ and } x \in O_{t,1} \\ 3 & \ \ \ \mbox{if } x \in I-O_{t,3} \text{ and } x \in O_{t,2} \\ \vdots & \ \ \ \ \ \ \ \ \ \ \vdots \\ j & \ \ \ \mbox{if } x \in I-O_{t,j} \text{ and } x \in O_{t,j-1} \\ \vdots & \ \ \ \ \ \ \ \ \ \ \vdots \end{cases}$

We are now ready to define $W=\{W_x: x \in I \}$ . For each $x \in I$ , $W_x$ is the mapping $W_x:I \rightarrow \omega$ defined by $W_x(t)=f_t(x)$ for each $t \in I$ .

Show the following:

The set $W=\{W_x: x \in I \}$ has cardinality continuum.
The set $W$ is a discrete space.
The set $W$ is a closed subspace of $Y$ .

Further Hints for Exercise 2.B

A subset $A$ of the real line $\mathbb{R}$ is nowhere dense in $\mathbb{R}$ if for any nonempty open subset $U$ of $\mathbb{R}$ , there is a nonempty open subset $V$ of $U$ such that $V \cap A=\varnothing$ . If we replace open sets by open intervals, we have the same notion.

Show that the real line $\mathbb{R}$ with the usual Euclidean topology cannot be the union of countably many closed and nowhere dense sets.

Further Hints for Exercise 2.C

Prove that if $X$ and $Y$ are $\sigma$ -compact, then the product $X \times Y$ is $\sigma$ -compact, hence Lindelof.

Prove that $S$ , the Sorgenfrey line, is Lindelof while its square $S \times S$ is not Lindelof.

Further Hints for Exercise 2.D

As suggested in the hints given earlier, prove that $X \times Y$ is Lindelof if $X$ is Lindelof and $Y$ is compact. As suggested, the Tube lemma is a useful tool.

Further Hints for Exercise 2.E

The product space $(0,1)^\omega$ is a subspace of the product space $[0,1]^\omega$ . Since $[0,1]^\omega$ is compact, we can fall back on a Baire category theorem argument to show why $(0,1)^\omega$ cannot be $\sigma$ -compact. To this end, we consider the notion of Baire space. A space $X$ is said to be a Baire space if for each countable family $\{ U_1,U_2,U_3,\cdots \}$ of open and dense subsets of $X$ , the intersection $\bigcap_{i=1}^\infty U_i$ is a dense subset of $X$ . Prove the following results.

Fact E.1
Let $X$ be a compact Hausdorff space. Let $O_1,O_2,O_3,\cdots$ be a sequence of non-empty open subsets of $X$ such that $\overline{O_{n+1}} \subset O_n$ for each $n$ . Then the intersection $\bigcap_{i=1}^\infty O_i$ is non-empty.

Fact E.2
Any compact Hausdorff space is Baire space.

Fact E.3
Let $X$ be a Baire space. Let $Y$ be a dense $G_\delta$ -subset of $X$ such that $X-Y$ is a dense subset of $X$ . Then $Y$ is not a $\sigma$ -compact space.

Since $X=[0,1]^\omega$ is compact, it follows from Fact E.2 that the product space $X=[0,1]^\omega$ is a Baire space.

Fact E.4
Let $X=[0,1]^\omega$ and $Y=(0,1)^\omega$ . The product space $Y=(0,1)^\omega$ is a dense $G_\delta$ -subset of $X=[0,1]^\omega$ . Furthermore, $X-Y$ is a dense subset of $X$ .

It follows from the above facts that the product space $(0,1)^\omega$ cannot be a $\sigma$ -compact space.

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Proofs of Key Steps for Exercise 2.A

The proof here focuses on the example $\mathbb{R}^c$ .

To see that $W=\{W_x: x \in I \}$ has the same cardinality as that of $I$ , show that $W_x \ne W_y$ for $x \ne y$ . This follows from the definition of the mapping $W_x$ .

To see that $W$ is discrete, for each $x \in I$ , consider the open set $U_x=\{ b \in Y: b(x)=0 \}$ . Note that $W_x \in U_x$ . Further note that $W_y \notin U_x$ for all $y \ne x$ .

To see that $W$ is a closed subset of $Y$ , let $k: I \rightarrow \omega$ such that $k \notin W$ . Consider two cases.

Case 1. $k(r) \ne 0$ for all $r \in I$ .
Note that $\{ O_{t,k(t)}: t \in I \}$ is an open cover of $I$ (in the usual topology). There exists a finite $H \subset I$ such that $\{ O_{h,k(h)}: h \in H \}$ is a cover of $I$ . Consider the open set $G=\{ b \in Y: \forall \ h \in H, \ b(h)=k(h) \}$ . Define the set $F$ as follows:

$F=\{ c \in I: W_c \in G \}$

The set $F$ can be further described as follows:

$\displaystyle \begin{aligned} F&=\{ c \in I: W_c \in G \} \\&=\{ c \in I: \forall \ h \in H, \ W_c(h)=f_h(c)=k(h) \ne 0 \} \\&=\{ c \in I: \forall \ h \in H, \ c \in I-O_{h,k(h)} \} \\&=\bigcap_{h \in H} (I-O_{h,k(h)}) \\&=I-\bigcup_{h \in H} O_{h,k(h)}=I-I =\varnothing \end{aligned}$

The last step is $\varnothing$ because $\{ O_{h,k(h)}: h \in H \}$ is a cover of $I$ . The fact that $F=\varnothing$ means that $G$ is an open subset of $Y$ containing the point $k$ such that $G$ contains no point of $W$ .

Case 2. $k(r) = 0$ for some $r \in I$ .
Since $k \notin W$ , $k \ne W_x$ for all $x \in I$ . In particular, $k \ne W_r$ . This means that $k(t) \ne W_r(t)$ for some $t \in I$ . Define the open set $G$ as follows:

$G=\{ b \in Y: b(r)=0 \text{ and } b(t)=k(t) \}$

Clearly $k \in G$ . Observe that $W_r \notin G$ since $W_r(t) \ne k(t)$ . For each $p \in I-\{ r \}$ , $W_p \notin G$ since $W_p(r) \ne 0$ . Thus $G$ is an open set containing $k$ such that $G \cap W=\varnothing$ .

Both cases show that $W$ is a closed subset of $Y=\omega^I$ .

Proofs of Key Steps for Exercise 2.B

Suppose that $\mathbb{P}$ , the set of all irrational numbers, is $\sigma$ -compact. That is, $\mathbb{P}=A_1 \cup A_2 \cup A_3 \cup \cdots$ where each $A_i$ is a compact space as a subspace of $\mathbb{P}$ . Any compact subspace of $\mathbb{P}$ is also a compact subspace of $\mathbb{R}$ . As a result, each $A_i$ is a closed subset of $\mathbb{R}$ . Furthermore, prove the following:

$A_i$

$\mathbb{R}$

Each singleton set $\{ r \}$ where $r$ is any rational number is also a closed and nowhere dense subset of $\mathbb{R}$ . This means that the real line is the union of countably many closed and nowhere dense subsets, contracting the hints given earlier. Thus $\mathbb{P}$ cannot be $\sigma$ -compact.

Proofs of Key Steps for Exercise 2.C

The Sorgenfrey line $S$ is a Lindelof space whose square $S \times S$ is not normal. This is a famous example of a Lindelof space whose square is not Lindelof (not even normal). For reference, a proof is found here. An alternative proof of the non-normality of $S \times S$ uses the Baire category theorem and is found here.

If the Sorgenfrey line is $\sigma$ -compact, then $S \times S$ would be $\sigma$ -compact and hence Lindelof. Thus $S$ cannot be $\sigma$ -compact.

Proofs of Key Steps for Exercise 2.D

Suppose that $X$ is Lindelof and that $Y$ is compact. Let $\mathcal{U}$ be an open cover of $X \times Y$ . For each $x \in X$ , let $\mathcal{U}_x \subset \mathcal{U}$ be finite such that $\mathcal{U}_x$ is a cover of $\{ x \} \times Y$ . Putting it another way, $\{ x \} \times Y \subset \cup \mathcal{U}_x$ . By the Tube lemma, for each $x \in X$ , there is an open $O_x$ such that $\{ x \} \times Y \subset O_x \times Y \subset \cup \mathcal{U}_x$ . Since $X$ is Lindelof, there exists a countable set $\{ x_1,x_2,x_3,\cdots \} \subset X$ such that $\{ O_{x_1},O_{x_2},O_{x_3},\cdots \}$ is a cover of $X$ . Then $\mathcal{U}_{x_1} \cup \mathcal{U}_{x_2} \cup \mathcal{U}_{x_3} \cup \cdots$ is a countable subcover of $\mathcal{U}$ . This completes the proof that $X \times Y$ is Lindelof when $X$ is Lindelof and $Y$ is compact.

To complete the exercise, observe that if $X$ is Lindelof and $Y$ is $\sigma$ -compact, then $X \times Y$ is the union of countably many Lindelof subspaces.

Proofs of Key Steps for Exercise 2.E

Proof of Fact E.1
Let $X$ be a compact Hausdorff space. Let $O_1,O_2,O_3,\cdots$ be a sequence of non-empty open subsets of $X$ such that $latex $\overline{O_{n+1}} \subset O_n$ for each $n$ . Show that the intersection $\bigcap_{i=1}^\infty O_i$ is non-empty.

Suppose that $\bigcap_{i=1}^\infty O_i=\varnothing$ . Choose $x_1 \in O_1$ . There must exist some $n_1$ such that $x_1 \notin O_{n_1}$ . Choose $x_2 \in O_{n_1}$ . There must exist some $n_2>n_1$ such that $x_2 \notin O_{n_2}$ . Continue in this manner we can choose inductively an infinite set $A=\{ x_1,x_2,x_3,\cdots \} \subset X$ such that $x_i \ne x_j$ for $i \ne j$ . Since $X$ is compact, the infinite set $A$ has a limit point $p$ . This means that every open set containing $p$ contains some $x_j$ (in fact for infinitely many $j$ ). The point $p$ cannot be in the intersection $\bigcap_{i=1}^\infty O_i$ . Thus for some $n$ , $p \notin O_n$ . Thus $p \notin \overline{O_{n+1}}$ . We can choose an open set $U$ such that $p \in U$ and $U \cap \overline{O_{n+1}}=\varnothing$ . However, $U$ must contain some point $x_j$ where $j>n+1$ . This is a contradiction since $O_j \subset \overline{O_{n+1}}$ for all $j>n+1$ . Thus Fact E.1 is established.

Proof of Fact E.2
Let $X$ be a compact space. Let $U_1,U_2,U_3,\cdots$ be open subsets of $X$ such that each $U_i$ is also a dense subset of $X$ . Let $V$ a non-empty open subset of $X$ . We wish to show that $V$ contains a point that belongs to each $U_i$ . Since $U_1$ is dense in $X$ , $O_1=V \cap U_1$ is non-empty. Since $U_2$ is dense in $X$ , choose non-empty open $O_2$ such that $\overline{O_2} \subset O_1$ and $O_2 \subset U_2$ . Since $U_3$ is dense in $X$ , choose non-empty open $O_3$ such that $\overline{O_3} \subset O_2$ and $O_3 \subset U_3$ . Continue inductively in this manner and we have a sequence of open sets $O_1,O_2,O_3,\cdots$ just like in Fact E.1. Then the intersection of the open sets $O_n$ is non-empty. Points in the intersection are in $V$ and in all the $U_n$ . This completes the proof of Fact E.2.

Proof of Fact E.3
Let $X$ be a Baire space. Let $Y$ be a dense $G_\delta$ -subset of $X$ such that $X-Y$ is a dense subset of $X$ . Show that $Y$ is not a $\sigma$ -compact space.

Suppose $Y$ is $\sigma$ -compact. Let $Y=\bigcup_{n=1}^\infty B_n$ where each $B_n$ is compact. Each $B_n$ is obviously a closed subset of $X$ . We claim that each $B_n$ is a closed nowhere dense subset of $X$ . To see this, let $U$ be a non-empty open subset of $X$ . Since $X-Y$ is dense in $X$ , $U$ contains a point $p$ where $p \notin Y$ . Since $p \notin B_n$ , there exists a non-empty open $V \subset U$ such that $V \cap B_n=\varnothing$ . This shows that each $B_n$ is a nowhere dense subset of $X$ .

Since $Y$ is a dense $G_\delta$ -subset of $X$ , $Y=\bigcap_{n=1}^\infty O_n$ where each $O_n$ is an open and dense subset of $X$ . Then each $A_n=X-O_n$ is a closed nowhere dense subset of $X$ . This means that $X$ is the union of countably many closed and nowhere dense subsets of $X$ . More specifically, we have the following.

(1)……… $X= \biggl( \bigcup_{n=1}^\infty A_n \biggr) \cup \biggl( \bigcup_{n=1}^\infty B_n \biggr)$

Statement (1) contradicts the fact that $X$ is a Baire space. Note that all $X-A_n$ and $X-B_n$ are open and dense subsets of $X$ . Further note that the intersection of all these countably many open and dense subsets of $X$ is empty according to (1). Thus $Y$ cannot not a $\sigma$ -compact space.

Proof of Fact E.4
The space $X=[0,1]^\omega$ is compact since it is a product of compact spaces. To see that $Y=(0,1)^\omega$ is a dense $G_\delta$ -subset of $X$ , note that $Y=\bigcap_{n=1}^\infty U_n$ where for each integer $n \ge 1$

(2)……… $U_n=(0,1) \times \cdots \times (0,1) \times [0,1] \times [0,1] \times \cdots$

Note that the first $n$ factors of $U_n$ are the open interval $(0,1)$ and the remaining factors are the closed interval $[0,1]$ . It is also clear that $X-Y$ is a dense subset of $X$ . This completes the proof of Fact E.4.

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Concluding Remarks

Exercise 2.A
The exercise is to show that the product of uncountably many $\sigma$ -compact spaces does not need to be Lindelof. The approach suggested in the hints is to show that $\mathbb{R}^{c}$ has uncountable extent where $c$ is continuum. Having uncountable extent (i.e. having an uncountable subset that is both closed and discrete) implies the space is not Lindelof. The uncountable extent of the product space $\mathbb{R}^{\omega_1}$ is discussed in this post.

For $\mathbb{R}^{c}$ and $\mathbb{R}^{\omega_1}$ , there is another way to show non-Lindelof. For example, both product spaces are not normal. As a result, both product spaces cannot be Lindelof. Note that every regular Lindelof space is normal. Both product spaces contain the product $\omega^{\omega_1}$ as a closed subspace. The non-normality of $\omega^{\omega_1}$ is discussed here.

Exercise 2.B
The hints given above is to show that the set of all irrational numbers, $\mathbb{P}$ , is not $\sigma$ -compact (as a subspace of the real line). The same argument showing that $\mathbb{P}$ is not $\sigma$ -compact can be generalized. Note that the complement of $\mathbb{P}$ is $\mathbb{Q}$ , the set of all rational numbers (a countable set). In this case, $\mathbb{Q}$ is a dense subset of the real line and is the union of countably many singleton sets. Each singleton set is a closed and nowhere dense subset of the real line. In general, we can let $B$ , the complement of a set $A$ , be dense in the real line and be the union of countably many closed nowhere dense subsets of the real line (not necessarily singleton sets). The same argument will show that $A$ cannot be a $\sigma$ -compact space. This argument is captured in Fact E.3 in Exercise 2.E. Thus both Exercise 2.B and Exercise 2.E use a Baire category argument.

Exercise 2.E
Like Exercise 2.B, this exercise is also to show a certain space is not $\sigma$ -compact. In this case, the suggested space is $\mathbb{R}^{\omega}$ , the product of countably many copies of the real line. The hints given use a Baire category argument, as outlined in Fact E.1 through Fact E.4. The product space $\mathbb{R}^{\omega}$ is embedded in the compact space $[0,1]^{\omega}$ , which is a Baire space. As mentioned earlier, Fact E.3 is essentially the same argument used for Exercise 2.B.

Using the same Baire category argument, it can be shown that $\omega^{\omega}$ , the product of countably many copies of the countably infinite discrete space, is not $\sigma$ -compact. The space $\omega$ of the non-negative integers, as a subspace of the real line, is certainly $\sigma$ -compact. Using the same Baire category argument, we can see that the product of countably many copies of this discrete space is not $\sigma$ -compact. With the product space $\omega^{\omega}$ , there is a connection with Exercise 2.B. The product $\omega^{\omega}$ is homeomorphic to $\mathbb{P}$ . The idea of the homeomorphism is discussed here. Thus the non- $\sigma$ -compactness of $\omega^{\omega}$ can be achieved by mapping it to the irrationals. Of course, the same Baire category argument runs through both exercises.

Exercise 2.C
Even the non- $\sigma$ -compactness of the Sorgenfrey line $S$ can be achieved by a Baire category argument. The non-normality of the Sorgenfrey plane $S \times S$ can be achieved by Jones’ lemma argument or by the fact that $\mathbb{P}$ is not a first category set. Links to both arguments are given in the Proof section above.

See here for another introduction to the Baire category theorem.

The Tube lemma is discussed here.

$\text{ }$

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$\copyright$ 2019 – Dan Ma

Lindelof Exercise 1

Posted on June 23, 2019 by Dan Ma

A space $X$ is called a $\sigma$ -compact space if it is the union of countably many compact subspaces. Clearly, any $\sigma$ -compact space is Lindelof. It is well known that the product of Lindelof spaces does not need to be Lindelof. The most well known example is perhaps the square of the Sorgenfrey line. In certain cases, the Lindelof property can be productive. For example, the product of countably many $\sigma$ -compact spaces is a Lindelof space. The discussion here centers on the following theorem.

Theorem 1
Let $X_1,X_2,X_3,\cdots$ be $\sigma$ -compact spaces. Then the product space $\prod_{i=1}^\infty X_i$ is Lindelof.

Theorem 1 is Exercise 3.8G in page 195 of General Topology by Engelking [1]. The reference for Exercise 3.8G is [2]. But the theorem is not found in [2] (it is not stated directly and it does not seem to be an obvious corollary of a theorem discussed in that paper). However, a hint is provided in Engelking for Exercise 3.8G. In this post, we discuss Theorem 1 as an exercise by giving expanded hint. Solutions to some of the key steps in the expanded hint are given at the end of the post.

Expanded Hint

It is helpful to first prove the following theorem.

Theorem 2
For each integer $i \ge 1$ , let $C_{i,1},C_{i,2},\cdots$ be compact spaces and let $C_i$ be the topological sum:

$C_i=C_{i,1} \oplus C_{i,2} \oplus C_{i,3} \oplus \cdots=\oplus_{j=1}^\infty C_{i,j}$

Then the product $\prod_{i=1}^\infty C_i$ is Lindelof.

Note that in the topological sum $C_{i,1} \oplus C_{i,2} \oplus C_{i,3} \oplus \cdots$ , the spaces $C_{i,1},C_{i,2},C_{i,3},\cdots$ are considered pairwise disjoint. The open sets in the sum are simply unions of the open sets in the individual spaces. Another way to view this topology: each of the $C_{i,j}$ is both closed and open in the topological sum. Theorem 2 is essentially saying that the product of countably many $\sigma$ -compact spaces is Lindelof if each $\sigma$ -compact space is the union of countably many disjoint compact spaces. The hint for Exercise 3.8G can be applied much more naturally on Theorem 2 than on Theorem 1. The following is Exercise 3.8F (a), which is the hint for Exercise 3.8G.

Lemma 3
Let $Z$ be a compact space. Let $X$ be a subspace of $Z$ . Suppose that there exist $F_1,F_2,F_3,\cdots$ , closed subsets of $Z$ , such that for all $x$ and $y$ where $x \in X$ and $y \in Z-X$ , there exists $F_i$ such that $x \in F_i$ and $y \notin F_i$ . Then $X$ is a Lindelof space.

The following theorem connects the hint (Lemma 3) with Theorem 2.

Theorem 4
For each integer $i \ge 1$ , let $Z_i$ be the one-point compactification of $C_i$ in Theorem 2. Then the product $Z=\prod_{i=1}^\infty Z_i$ is a compact space. Furthermore, $X=\prod_{i=1}^\infty C_i$ is a subspace of $Z$ . Prove that $Z$ and $X$ satisfy Lemma 3.

Each $C_i$ in Theorem 2 is a locally compact space. To define the one-point compactifications, for each $i$ , choose $p_i \notin C_i$ . Make sure that $p_i \ne p_j$ for $i \ne j$ . Then $Z_i$ is simply

$Z_i=C_i \cup \{ p_i \}=C_{i,1} \oplus C_{i,2} \oplus C_{i,3} \oplus \cdots \cup \{ p_i \}$

with the topology defined as follows:

Open subsets of $C_i$ continue to be open in $Z_i$ .
An open set containing $p_i$ is of the form $\{ p_i \} \cup (C_i - \overline{D})$ where $D$ is open in $C_i$ and $D$ is contained in the union of finitely many $C_{i,j}$ .

For convenience, each point $p_i$ is called a point at infinity.

Note that Theorem 2 follows from Lemma 3 and Theorem 4. In order to establish Theorem 1 from Theorem 2, observe that the Lindelof property is preserved by any continuous mapping and that there is a natural continuous map from the product space in Theorem 2 to the product space in Theorem 1.

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Proofs of Key Steps

Proof of Lemma 3
Let $Z$ , $X$ and $F_1,F_2,F_3,\cdots$ be as described in the statement for Lemma 3. Let $\mathcal{U}$ be a collection of open subsets of $Z$ such that $\mathcal{U}$ covers $X$ . We would like to show that a countable subcollection of $\mathcal{U}$ is also a cover of $X$ . Let $O=\cup \mathcal{U}$ . If $Z-O=\varnothing$ , then $\mathcal{U}$ is an open cover of $Z$ and there is a finite subset of $\mathcal{U}$ that is a cover of $Z$ and thus a cover of $X$ . Thus we can assume that $Z-O \ne \varnothing$ .

Let $F=\{ F_1,F_2,F_3,\cdots \}$ . Let $K=Z-O$ , which is compact. We make the following claim.

Claim. Let $Y$ be the union of all possible $\cap G$ where $G \subset F$ is finite and $\cap G \subset O$ . Then $X \subset Y \subset O$ .

To establish the claim, let $x \in X$ . For each $y \in K=Z-O$ , there exists $F_{n(y)}$ such that $x \in F_{n(y)}$ and $y \notin F_{n(y)}$ . This means that $\{ Z-F_{n(y)}: y \in K \}$ is an open cover of $K$ . By the compactness of $K$ , there are finitely many $F_{n(y_1)}, \cdots, F_{n(y_k)}$ such that $F_{n(y_1)} \cap \cdots \cap F_{n(y_k)}$ misses $K$ , or equivalently $F_{n(y_1)} \cap \cdots \cap F_{n(y_k)} \subset O$ . Note that $x \in F_{n(y_1)} \cap \cdots \cap F_{n(y_k)}$ . Further note that $F_{n(y_1)} \cap \cdots \cap F_{n(y_k)} \subset Y$ . This establishes the claim that $X \subset Y$ . The claim that $Y \subset O$ is clear from the definition of $Y$ .

Each set $F_i$ is compact since it is closed in $Z$ . The intersection of finitely many $F_i$ is also compact. Thus the $\cap G$ in the definition of $Y$ in the above claim is compact. There can be only countably many $\cap G$ in the definition of $Y$ . Thus $Y$ is a $\sigma$ -compact space that is covered by the open cover $\mathcal{U}$ . Choose a countable $\mathcal{V} \subset \mathcal{U}$ such that $\mathcal{V}$ covers $Y$ . Then $\mathcal{V}$ is a cover of $X$ too. This completes the proof that $X$ is Lindelof.

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Proof of Theorem 4
Recall that $Z=\prod_{i=1}^\infty Z_i$ and that $X=\prod_{i=1}^\infty C_i$ . Each $Z_i$ is the one-point compactification of $C_i$ , which is the topological sum of the disjoint compact spaces $C_{i,1},C_{i,2},\cdots$ .

For integers $i,j \ge 1$ , define $K_{i,j}=C_{i,1} \oplus C_{i,2} \oplus \cdots \oplus C_{i,j}$ . For integers $n,j \ge 1$ , define the product $F_{n,j}$ as follows:

$F_{n,j}=K_{1,j} \times \cdots \times K_{n,j} \times Z_{n+1} \times Z_{n+2} \times \cdots$

Since $F_{n,j}$ is a product of compact spaces, $F_{n,j}$ is compact and thus closed in $Z$ . There are only countably many $F_{n,j}$ .

We claim that the countably many $F_{n,j}$ have the property indicated in Lemma 3. To this end, let $f \in X=\prod_{i=1}^\infty C_i$ and $g \in Z-X$ . There exists an integer $n \ge 1$ such that $g(n) \notin C_{n}$ . This means that $g(n) \notin C_{n,j}$ for all $j$ , i.e. $g(n)=p_n$ (so $g(n)$ must be the point at infinity). Choose $j \ge 1$ large enough such that

$f(i) \in K_{i,j}=C_{i,1} \oplus C_{i,2} \oplus \cdots \oplus C_{i,j}$

for all $i \le n$ . It follows that $f \in F_{n,j}$ and $g \notin F_{n,j}$ . Thus the sequence of closed sets $F_{n,j}$ satisfies Lemma 3. By Lemma 3, $X=\prod_{i=1}^\infty C_i$ is Lindelof.

Reference

Engelking R., General Topology, Revised and Completed edition, Elsevier Science Publishers B. V., Heldermann Verlag, Berlin, 1989.
Hager A. W., Approximation of real continuous functions on Lindelof spaces, Proc. Amer. Math. Soc., 22, 156-163, 1969.

$\text{ }$

Dan Ma topology

Daniel Ma topology

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$\copyright$ 2019 – Dan Ma

The product of locally compact paracompact spaces

Posted on April 1, 2017 by Dan Ma

It is well known that when $X$ and $Y$ are paracompact spaces, the product space $X \times Y$ is not necessarily normal. Classic examples include the product of the Sorgenfrey line with itself (discussed here) and the product of the Michael line and the space of irrational numbers (discussed here). However, if one of the paracompact factors is “compact”, the product can be normal or even paracompact. This post discusses several classic results along this line. All spaces are Hausdorff and regular.

Suppose that $X$ and $Y$ are paracompact spaces. We have the following results:

If $Y$ is a compact space, then $X \times Y$ is paracompact.
If $Y$ is a $\sigma$ -compact space, then $X \times Y$ is paracompact.
If $Y$ is a locally compact space, then $X \times Y$ is paracompact.
If $Y$ is a $\sigma$ -locally compact space, then $X \times Y$ is paracompact.

The proof of the first result makes uses the tube lemma. The second result is a corollary of the first. The proofs of both results are given here. The third result is a corollary of the fourth result. We give a proof of the fourth result.

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Proof of the Fourth Result

The fourth result indicated above is restated as Theorem 2 below. It is a theorem of K. Morita [1]. This is one classic result on product of paracompact spaces. After proving the theorem, comments are made about interesting facts and properties that follow from this result. Theorem 2 is also Theorem 3.22 in chapter 18 in the Handbook of Set-Theoretic Topology [2].

A space $W$ is a locally compact space if for each $w \in W$ , there is an open subset $O$ of $W$ such that $w \in O$ and $\overline{O}$ is compact. When we say $Y$ is a $\sigma$ -locally compact space, we mean that $Y=\bigcup_{j=1}^\infty Y_j$ where each $Y_j$ is a locally compact space. In proving the result discussed here, we also assume that each $Y_j$ is a closed subspace of $Y$ . The following lemma will be helpful.

Lemma 1
Let $Y$ be a paracompact space. Suppose that $Y$ is $\sigma$ -locally compact. Then there exists a cover $\mathcal{C}=\bigcup_{j=1}^\infty \mathcal{C}_j$ of $Y$ such that each $\mathcal{C}_j$ is a locally finite family consisting of compact sets.

Proof of Lemma 1
Let $Y=\bigcup_{n=1}^\infty Y_n$ such that each $Y_n$ is closed and is locally compact. Fix an integer $n$ . For each $y \in Y_n$ , let $O_{n,y}$ be an open subset of $Y_n$ such that $y \in O_{n,y}$ and $\overline{O_{n,y}}$ is compact (the closure is taken in $Y_n$ ). Consider the open cover $\mathcal{O}=\left\{ O_{n,y}: y \in Y_j \right\}$ of $Y_n$ . Since $Y_n$ is a closed subspace of $Y$ , $Y_n$ is also paracompact. Let $\mathcal{V}=\left\{ V_{n,y}: y \in Y_j \right\}$ be a locally finite open cover of $Y_n$ such that $\overline{V_{n,y}} \subset O_{n,y}$ for each $y \in Y_n$ (again the closure is taken in $Y_n$ ). Each $\overline{V_{n,y}}$ is compact since $\overline{V_{n,y}} \subset O_{n,y} \subset \overline{O_{n,y}}$ . Let $\mathcal{C}_n=\left\{ \overline{V_{n,y}}: y \in Y_n \right\}$ .

We claim that $\mathcal{C}_n$ is a locally finite family with respect to the space $Y$ . For each $y \in Y-Y_n$ , $Y-Y_n$ is an open set containing $y$ that intersects no set in $\mathcal{C}_n$ . For each $y \in Y_n$ , there is an open set $O \subset Y_n$ that meets only finitely many sets in $\mathcal{C}_n$ . Extend $O$ to an open subset $O_1$ of $Y$ . That is, $O_1$ is an open subset of $Y$ such that $O=O_1 \cap Y_n$ . It is clear that $O_1$ can only meets finitely many sets in $\mathcal{C}_n$ .

Then $\mathcal{C}=\bigcup_{j=1}^\infty \mathcal{C}_j$ is the desired $\sigma$ -locally finite cover of $Y$ . $\square$

Theorem 2
Let $X$ be any paracompact space and let $Y$ be any $\sigma$ -locally compact paracompact space. Then $X \times Y$ is paracompact.

Proof of Theorem 2
By Lemma 1, let $\mathcal{C}=\bigcup_{n=1}^\infty \mathcal{C}_n$ be a $\sigma$ -locally finite cover of $Y$ such that each $\mathcal{C}_n$ consists of compact sets. To show that $X \times Y$ is paracompact, let $\mathcal{U}$ be an open cover of $X \times Y$ . For each $C \in \mathcal{C}$ and for each $x \in X$ , the set $\left\{ x \right\} \times C$ is obviously compact.

Fix $C \in \mathcal{C}$ and fix $x \in X$ . For each $y \in C$ , the point $(x,y) \in U_{y}$ for some $U_{y} \in \mathcal{U}$ . Choose open $H_y \subset X$ and open $K_y \subset Y$ such that $(x,y) \in H_y \times K_y \subset U_{x,y}$ . Letting $y$ vary, the open sets $H_y \times K_y$ cover the compact set $\left\{ x \right\} \times C$ . Choose finitely many open sets $H_y \times K_y$ that also cover $\left\{ x \right\} \times C$ . Let $H(C,x)$ be the intersection of these finitely many $H_y$ . Let $\mathcal{K}(C,x)$ be the set of these finitely many $K_y$ .

To summarize what we have obtained in the previous paragraph, for each $C \in \mathcal{C}$ and for each $x \in X$ , there exists an open subset $H(C,x)$ containing $x$ , and there exists a finite set $\mathcal{K}(C,x)$ of open subsets of $Y$ such that

$C \subset \bigcup \mathcal{K}(C,x)$ ,
for each $K \in \mathcal{K}(C,x)$ , $H(C,x) \times K \subset U$ for some $U \in \mathcal{U}$ .

For each $C \in \mathcal{C}$ , the set of all $H(C,x)$ is an open cover of $X$ . Since $X$ is paracompact, for each $C \in \mathcal{C}$ , there exists a locally finite open cover $\mathcal{L}_C=\left\{L(C,x): x \in X \right\}$ such that $L(C,x) \subset H(C,x)$ for all $x$ . Consider the following families of open sets.

$\mathcal{E}_n=\left\{L(C,x) \times K: C \in \mathcal{C}_n \text{ and } x \in X \text{ and } K \in \mathcal{K}(C,x) \right\}$

$\mathcal{E}=\bigcup_{n=1}^\infty \mathcal{E}_n$

We claim that $\mathcal{E}$ is a $\sigma$ -locally finite open refinement of $\mathcal{U}$ . First, show that $\mathcal{E}$ is an open cover of $X \times Y$ . Let $(a,b) \in X \times Y$ . Then for some $n$ , $b \in C$ for some $C \in \mathcal{C}_n$ . Furthermore, $a \in L(C,x)$ for some $x \in X$ . The information about $C$ and $x$ are detailed above. For example, $C \subset \bigcup \mathcal{K}(C,x)$ . Thus there exists some $K \in \mathcal{K}(C,x)$ such that $b \in K$ . We now have $(a,b) \in L(C,x) \times K \in \mathcal{E}_n$ .

Next we show that $\mathcal{E}$ is a refinement of $\mathcal{U}$ . Fix $L(C,x) \times K \in \mathcal{E}_n$ . Immediately we see that $L(C,x) \subset H(C,x)$ . Since $K \in \mathcal{K}(C,x)$ , $H(C,x) \times K \subset U$ for some $U \in \mathcal{U}$ . Then $L(C,x) \times K \subset U$ .

The remaining point to make is that each $\mathcal{E}_n$ is a locally finite family of open subsets of $X \times Y$ . Let $(a,b) \in X \times Y$ . Since $\mathcal{C}_n$ is locally finite in $Y$ , there exists some open $Q \subset Y$ such that $b \in Q$ and $Q$ meets only finitely many sets in $\mathcal{C}_n$ , say $C_1,C_2,\cdots,C_m$ . Recall that $\mathcal{L}_{C_j}$ is the set of all $L(C_j,x)$ and is locally finite. Thus there exists an open $O \subset X$ such that $a \in O$ and $O$ meets only finitely many sets in each $\mathcal{L}_{C_j}$ where $j=1,2,\cdots,m$ . Thus the open set $O$ meets only finitely many sets $L(C,x)$ for finitely many $C \in \mathcal{C}_n$ and finitely many $x \in X$ . These finitely many $C$ and $x$ lead to finitely many $K$ . Thus it follows that $O \times Q$ meets only finitely many sets $L(C,x) \times K$ in $\mathcal{E}_n$ . Thus $\mathcal{E}_n$ is locally finite.

What has been established is that every open cover of $X \times Y$ has a $\sigma$ -locally finite open refinement. This fact is equivalent to paracompactness (according to Theorem 1 in this previous post). This concludes the proof of the theorem. $\square$

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Productively Paracompact Spaces

Consider this property for a space $X$ .

$X$

$X \times Y$

$Y$

Such a space can be called a productively paracompact space (for some reason, this term is not used in the literature).

According to the four results stated at the beginning, any space in any one of the following four classes

Compact spaces.
$\sigma$ -compact spaces.
Locally compact paracompact spaces.
$\sigma$ -locally compact paracompact spaces.

satisfies this property. Both the Michael line and the space of the irrational numbers are examples of paracompact spaces that do not have this productively paracompact property. According to comments made on page 799 [2], the theorem of Morita (Theorem 2 here) triggered extensive research to investigate this class of spaces. The class of spaces is broader than the four classes listed here. For example, the productively paracompact spaces also include the closed images of locally compact paracompact spaces. The handbook [2] has more references.

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Normal P-Spaces

Consider this property.

$X$

$X \times Y$

$Y$

These spaces can be called productively normal spaces with respect to metric spaces. They go by another name. Morita defined the notion of P-spaces and proved that a space $X$ is a normal P-space if and only if the product of $X$ with any metric space is normal.

Since the class of metric spaces contain the paracompact spaces, any space has property (*) would have property (**), i.e. a normal P-space.Thus any locally compact paracompact space is a normal P-space. Any $\sigma$ -locally compact paracompact space is a normal P-space. If a paracompact space has any one of the four “compact” properties discussed here, it is a normal P-space.

Other examples of normal P-spaces are countably compact normal spaces (see here) and perfectly normal spaces (see here).

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Looking at Diagrams

Let’s compare these classes of spaces: productively paracompact spaces (the spaces satisfying property (*)), normal P-spaces and paracompact spaces. We have the following diagram.

Diagram 1

Clearly productively paracompact implies paracompact. As discussed in the previous section, productively paracompact implies normal P. If a space $X$ is such that the product of $X$ with every paracompact space is paracompact, then the product of $X$ with every metric space is paracompact and hence normal.

However, the arrows in Diagram 1 are not reversible. The Michael line mentioned at the beginning will shed some light on this point. Here’s the previous post on Michael line. Let $\mathbb{M}$ be the Michael line. Let $\mathbb{P}$ be the space of the irrational numbers. The space $\mathbb{M}$ would be a paracompact space that is not productively paracompact since its product with $\mathbb{P}$ is not normal, hence not paracompact.

On the other hand, the space of irrational numbers $\mathbb{P}$ is a normal P-space since it is a metric space. But it is not productively paracompact since its product with the Michael line $\mathbb{M}$ is not normal, hence not paracompact.

The two classes of spaces at the bottom of Diagram 1 do not relate. The Michael line $\mathbb{M}$ is a paracompact space that is not a normal P-space since its product with $\mathbb{P}$ is not normal. Normal P-space does not imply paracompact. Any space that is normal and countably compact is a normal P-space. For example, the space $\omega_1$ , the first uncountable ordinal, with the ordered topology is normal and countably compact and is not paracompact.

There are other normal P-spaces that are not paracompact. For example, Bing’s Example H is perfectly normal and not paracompact. As mentioned in the previous section, any perfectly normal space is a normal P-space.

The class of spaces whose product with every paracompact space is paracompact is stronger than both classes of paracompact spaces and normal P-spaces. It is a strong property and an interesting class of spaces. It is also an excellent topics for any student who wants to dig deeper into paracompact spaces.

Let’s add one more property to Diagram 1.

Diagram 2

$\displaystyle \begin{array}{ccccc} \text{ } &\text{ } & \text{Productively Paracompact} & \text{ } & \text{ } \\ \text{ } & \swarrow & \text{ } & \searrow & \text{ } \\ \text{Paracompact} &\text{ } & \text{ } & \text{ } & \text{Normal P-space} \\ \text{ } & \searrow & \text{ } & \swarrow & \text{ } \\ \text{ } &\text{ } & \text{Normal Countably Paracompact} & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \end{array}$

All properties in Diagram 2 except for paracompact are productive. Normal countably paracompact spaces are productive. According to Dowker’s theorem, the product of any normal countably paracompact space with any compact metric space is normal (see Theorem 1 in this previous post). The last two arrows in Diagram 2 are also not reversible.

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Reference

Morita K., On the Product of Paracompact Spaces, Proc. Japan Acad., Vol. 39, 559-563, 1963.
Przymusinski T. C., Products of Normal Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 781-826, 1984.

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$\copyright$ 2017 – Dan Ma

The product of a perfectly normal space and a metric space is perfectly normal

Posted on March 1, 2017 by Dan Ma

The previous post gives a positive result for normality in product space. It shows that the product of a normal countably compact space and a metric space is always normal. In this post, we discuss another positive result, which is the following theorem.

Main Theorem
If $X$ is a perfectly normal space and $Y$ is a metric space, then $X \times Y$ is a perfectly normal space.

As a result of this theorem, perfectly normal spaces belong to a special class of spaces called P-spaces. K. Morita defined the notion of P-space and he proved that a space $Y$ is a Normal P-space if and only if $X \times Y$ is normal for every metric space $X$ (see the section below on P-spaces). Thus any perfectly normal space is a Normal P-space.

All spaces under consideration are Hausdorff. A subset $A$ of the space $X$ is a $G_\delta$ -subset of the space $X$ if $A$ is the intersection of countably many open subsets of $X$ . A subset $B$ of the space $X$ is an $F_\sigma$ -subset of the space $X$ if $B$ is the union of countably many closed subsets of $X$ . Clearly, a set $A$ is a $G_\delta$ -subset of the space $X$ if and only if $X-A$ is an $F_\sigma$ -subset of the space $X$ .

A space $X$ is said to be a perfectly normal space if $X$ is normal with the additional property that every closed subset of $X$ is a $G_\delta$ -subset of $X$ (or equivalently every open subset of $X$ is an $F_\sigma$ -subset of $X$ ).

The perfect normality has a characterization in terms of zero-sets and cozero-sets. A subset $A$ of the space $X$ is said to be a zero-set if there exists a continuous function $f: X \rightarrow [0,1]$ such that $A=f^{-1}(0)$ , where $f^{-1}(0)=\left\{x \in X: f(x)=0 \right\}$ . A subset $B$ of the space $X$ is a cozero-set if $X-B$ is a zero-set, or more explicitly if there is a continuous function $f: X \rightarrow [0,1]$ such that $B=\left\{x \in X: f(x)>0 \right\}$ .

It is well known that the space $X$ is perfectly normal if and only if every closed subset of $X$ is a zero-set, equivalently every open subset of $X$ is a cozero-set. See here for a proof of this result. We use this result to show that $X \times Y$ is perfectly normal.

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The Proof

Let $X$ be a perfectly normal space and $Y$ be a metric space. Since $Y$ is a metric space, let $\mathcal{B}=\bigcup_{j=1}^\infty \mathcal{B}_j$ be a base for $Y$ such that each $\mathcal{B}_j$ is locally finite. We show that $X \times Y$ is perfectly normal. To that end, we show that every open subset of $X \times Y$ is a cozero-set. Let $U$ be an open subset of $X \times Y$ .

For each $(x,y) \in X \times Y$ , there exists open $O_{x,y} \subset X$ and there exists $B_{x,y} \in \mathcal{B}$ such that $(x,y) \in O_{x,y} \times B_{x,y} \subset U$ . Then $U$ is the union of all sets $O_{x,y} \times B_{x,y}$ . Observe that $B_{x,y} \in \mathcal{B}_{j}$ for some integer $j$ . For each $B \in \mathcal{B}$ such that $B=B_{x,y}$ for some $(x,y) \in X \times Y$ , let $O(B)$ be the union of all corresponding open sets $O_{x,y}$ for all applicable $(x,y)$ .

For each positive integer $j$ , let $\mathcal{W}_j$ be the collection of all open sets $O(B) \times B$ such that $B \in \mathcal{B}_j$ and $B=B_{x,y}$ for some $(x,y) \in X \times Y$ . Let $\mathcal{V}_j=\cup \mathcal{W}_j$ . As a result, $U=\bigcup_{j=1}^\infty \mathcal{V}_j$ .

Since both $X$ and $Y$ are perfectly normal, for each $O(B) \times B \in \mathcal{W}_j$ , there exist continuous functions

$F_{O(B),j}: X \rightarrow [0,1]$

$G_{B,j}: Y \rightarrow [0,1]$

such that

$O(B)=\left\{x \in X: F_{O(B),j}(x) >0 \right\}$

$B=\left\{y \in Y: G_{B,j}(y) >0 \right\}$

Now define $H_j: X \times Y \rightarrow [0,1]$ by the following:

$\displaystyle H_j(x,y)=\sum \limits_{O(B) \times B \in \mathcal{W}_j} F_{O(B),j}(x) \ G_{B,j}(y)$

for all $(x,y) \in X \times Y$ . Note that the function $H_j$ is well defined. Since $\mathcal{B}_j$ is locally finite in $Y$ , $\mathcal{W}_j$ is locally finite in $X \times Y$ . Thus $H_j(x,y)$ is obtained by summing a finite number of values of $F_{O(B),j}(x) \ G_{B,j}(y)$ . On the other hand, it can be shown that $H_j$ is continuous for each $j$ . Based on the definition of $H_j$ , it can be readily verified that $H_j(x,y)>0$ for all $(x,y) \in \cup \mathcal{W}_j$ and $H_j(x,y)=0$ for all $(x,y) \notin \cup \mathcal{W}_j$ .

Define $H: X \times Y \rightarrow [0,1]$ by the following:

$\displaystyle H(x,y)=\sum \limits_{j=1}^\infty \biggl[ \frac{1}{2^j} \ \frac{H_j(x,y)}{1+H_j(x,y)} \biggr]$

It is clear that $H$ is continuous. We claim that $U=\left\{(x,y) \in X \times Y: H(x,y) >0 \right\}$ . Recall that the open set $U$ is the union of all $O(B) \times B \in \mathcal{W}_j$ for all $j$ . Thus if $(x,y) \in \cup \mathcal{W}_j$ for some $j$ , then $H(x,y)>0$ since $H_j(x,y)>0$ . If $(x,y) \notin \cup \mathcal{W}_j$ for all $j$ , $H(x,y)=0$ since $H_j(x,y)=0$ for all $j$ . Thus the open set $U$ is an $F_\sigma$ -subset of $X \times Y$ . This concludes the proof that $X \times Y$ is perfectly normal. $\square$

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Remarks

The main theorem here is a classic result in general topology. An alternative proof is to show that any perfectly normal space is a P-space (definition given below). Then by Morita’s theorem, the product of any perfectly normal space and any metric space is normal (Theorem 1 below). For another proof that is elementary, see Lemma 7 in this previous post.

The notions of perfectly normal spaces and paracompact spaces are quite different. By the theorem discussed here, perfectly normal spaces are normally productive with metric spaces. It is possible for a paracompact space to have a non-normal product with a metric space. The classic example is the Michael line (discussed here).

On the other hand, there are perfectly normal spaces that are not paracompact. One example is Bing’s Example H, which is perfectly normal and not paracompact (see here).

Even though a perfectly normal space is normally productive with metric spaces, it cannot be normally productive in general. For each non-discrete perfectly normal space $X$ , there exists a normal space $Y$ such that $X \times Y$ is not normal. This follows from Morita’s first conjecture (now a true statement). Morita’s first conjecture is discussed here.

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P-Space in the Sense of Morita

Morita defined the notion of P-spaces [1] and [2]. Let $\kappa$ be a cardinal number such that $\kappa \ge 1$ . Let $\Gamma$ be the set of all finite ordered sequences $(\alpha_1,\alpha_2,\cdots,\alpha_n)$ where $n=1,2,\cdots$ and all $\alpha_i < \kappa$ . Let $X$ be a space. The collection $\left\{F_\sigma \subset X: \sigma \in \Gamma \right\}$ is said to be decreasing if this condition holds: $\sigma =(\alpha_1,\alpha_2,\cdots,\alpha_n)$ and $\delta =(\alpha_1,\alpha_2,\cdots,\alpha_n, \cdots, \alpha_m)$ with $n<m$ imply that $F_{\delta} \subset F_{\sigma}$ . The space $X$ is a P-space if for any cardinal $\kappa \ge 1$ and for any decreasing collection $\left\{F_\sigma \subset X: \sigma \in \Gamma \right\}$ of closed subsets of $X$ , there exists open set $U_\sigma$ for each $\sigma \in \Gamma$ such that the following conditions hold:

for all $\sigma \in \Gamma$ , $F_\sigma \subset U_\sigma$ ,
for any infinite sequence $(\alpha_1,\alpha_2,\cdots,\alpha_n,\cdots)$ where each each finite subsequence $\sigma_n=(\alpha_1,\alpha_2,\cdots,\alpha_n)$ is an element of $\Gamma$ , if $\bigcap_{n=1}^\infty F_{\sigma_n}=\varnothing$ , then $\bigcap_{n=1}^\infty U_{\sigma_n}=\varnothing$ .

If $\kappa=1$ where $1=\left\{0 \right\}$ . Then the index set $\Gamma$ defined above can be viewed as the set of all positive integers. As a result, the definition of P-space with $\kappa=1$ implies the a condition in Dowker’s theorem (see condition 6 in Theorem 1 here). Thus any space $X$ that is normal and a P-space is countably paracompact (or countably shrinking or that $X \times Y$ is normal for every compact metric space or any other equivalent condition in Dowker’s theorem). The following is a theorem of Morita.

Theorem 1 (Morita)
Let $X$ be a space. Then $X$ is a normal P-space if and only if $X \times Y$ is normal for every metric space $Y$ .

In light of Theorem 1, both perfectly normal spaces and normal countably compact spaces are P-spaces (see here). According to Theorem 1 and Dowker’s theorem, it follows that any normal P-space is countably paracompact.

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Reference

Morita K., On the Product of a Normal Space with a Metric Space, Proc. Japan Acad., Vol. 39, 148-150, 1963. (article information; paper)
Morita K., Products of Normal Spaces with Metric Spaces, Math. Ann., Vol. 154, 365-382, 1964.

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$\copyright \ 2017 \text{ by Dan Ma}$

The product of a normal countably compact space and a metric space is normal

Posted on February 5, 2017 by Dan Ma

It is well known that normality is not preserved by taking products. When nothing is known about the spaces $X$ and $Y$ other than the facts that they are normal spaces, there is not enough to go on for determining whether $X \times Y$ is normal. In fact even when one factor is a metric space and the other factor is a hereditarily paracompact space, the product can be non-normal (discussed here). This post discusses a productive scenario – the first factor is a normal space and second factor is a metric space with the first factor having the additional property that it is countably compact. In this scenario the product is always normal. This is a well known result in general topology. The goal here is to nail down a proof for use as future reference.

Main Theorem
Let $X$ be a normal and countably compact space. Then $X \times Y$ is a normal space for every metric space $Y$ .

The proof of the main theorem uses the notion of shrinkable open covers.

Remarks
The main theorem is a classic result and is often used as motivation for more advanced results for products of normal spaces. Thus we would like to present a clear and complete proof of this classic result for anyone who would like to study the topics of normality (or the lack of) in product spaces. We found that some proofs of this result in the literature are hard to follow. In A. H. Stone’s paper [2], the result is stated in a footnote, stating that “it can be shown that the topological product of a metric space and a normal countably compact space is normal, though not necessarily paracompact”. We had seen several other papers citing [2] as a reference for the result. The Handbook [1] also has a proof (Corollary 4.10 in page 805), which we feel may not be the best proof to learn from. We found a good proof in [3] using the idea of shrinking of open covers.

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The Notion of Shrinking

The key to the proof is the notion of shrinkable open covers and shrinking spaces. Let $X$ be a space. Let $\mathcal{U}$ be an open cover of $X$ . The open cover of $\mathcal{U}$ is said to be shrinkable if there is an open cover $\mathcal{V}=\left\{V(U): U \in \mathcal{U} \right\}$ of $X$ such that $\overline{V(U)} \subset U$ for each $U \in \mathcal{U}$ . When this is the case, the open cover $\mathcal{V}$ is said to be a shrinking of $\mathcal{U}$ . If an open cover is shrinkable, we also say that the open cover can be shrunk (or has a shrinking). Whenever an open cover has a shrinking, the shrinking is indexed by the open cover that is being shrunk. Thus if the original cover is indexed, e.g. $\left\{U_\alpha: \alpha<\kappa \right\}$ , then a shrinking has the same indexing, e.g. $\left\{V_\alpha: \alpha<\kappa \right\}$ .

A space $X$ is a shrinking space if every open cover of $X$ is shrinkable. Every open cover of a paracompact space has a locally finite open refinement. With a little bit of rearranging, the locally finite open refinement can be made to be a shrinking (see Theorem 2 here). Thus every paracompact space is a shrinking space. For other spaces, the shrinking phenomenon is limited to certain types of open covers. In a normal space, every finite open cover has a shrinking, as stated in the following theorem.

Theorem 1
The following conditions are equivalent.

The space $X$ is normal.
Every point-finite open cover of $X$ is shrinkable.
Every locally finite open cover of $X$ is shrinkable.
Every finite open cover of $X$ is shrinkable.
Every two-element open cover of $X$ is shrinkable.

The hardest direction in the proof is $1 \Longrightarrow 2$ , which is established in this previous post. The directions $2 \Longrightarrow 3 \Longrightarrow 4 \Longrightarrow 5$ are immediate. To see $5 \Longrightarrow 1$ , let $H$ and $K$ be two disjoint closed subsets of $X$ . By condition 5, the two-element open cover $\left\{X-H,X-K \right\}$ has a shrinking $\left\{U,V \right\}$ . Then $\overline{U} \subset X-H$ and $\overline{V} \subset X-K$ . As a result, $H \subset X-\overline{U}$ and $K \subset X-\overline{V}$ . Since the open sets $U$ and $V$ cover the whole space, $X-\overline{U}$ and $X-\overline{V}$ are disjoint open sets. Thus $X$ is normal.

In a normal space, all finite open covers are shrinkable. In general, an infinite open cover of a normal space may or may not be shrinkable. It turns out that finding a normal space with an infinite open cover that is not shrinkable is no trivial matter (see Dowker’s theorem in this previous post). However, if an open cover in a normal space is point-finite or locally finite, then it is shrinkable.

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Key Idea

We now discuss the key idea to the proof of the main theorem. Consider the product space $X \times Y$ . Let $\mathcal{U}$ be an open cover of $X \times Y$ . Let $M \subset Y$ . The set $M$ is stable with respect to the open cover $\mathcal{U}$ if for each $x \in X$ , there is an open set $O_x$ containing $x$ such that $O_x \times M \subset U$ for some $U \in \mathcal{U}$ .

Let $\kappa$ be a cardinal number (either finite or infinite). A space $X$ is a $\kappa$ -shrinking space if for each open cover $\mathcal{W}$ of $X$ such that the cardinality of $\mathcal{W}$ is $\le \kappa$ , then $\mathcal{W}$ is shrinkable. According to Theorem 1, any normal space is 2-shrinkable.

Theorem 2
Let $\kappa$ be a cardinal number (either finite or infinite). Let $X$ be a $\kappa$ -shrinking space. Let $Y$ be a paracompact space. Suppose that $\mathcal{U}$ is an open cover of $X \times Y$ such that the following two conditions are satisfied:

Each point $y \in Y$ has an open set $V_y$ containing $y$ such that $V_y$ is stable with respect to $\mathcal{U}$ .
$\lvert \mathcal{U} \lvert = \kappa$ .

Then $\mathcal{U}$ is shrinkable.

Proof of Theorem 2
Let $\mathcal{U}$ be any open cover of $X \times Y$ satisfying the hypothesis. We show that $\mathcal{U}$ has a shrinking.

For each $y \in Y$ , obtain the open covers $\left\{G(U,y): U \in \mathcal{U} \right\}$ and $\left\{H(U,y): U \in \mathcal{U} \right\}$ of $X$ as follows. For each $U \in \mathcal{U}$ , define the following:

$G(U,y)=\cup \left\{O: O \text{ is open in } X \text{ such that } O \times V_y \subset U \right\}$

Then $\left\{G(U,y): U \in \mathcal{U} \right\}$ is an open cover of $X$ . Since $X$ is $\kappa$ -shrinkable, there is an open cover $\left\{H(U,y): U \in \mathcal{U} \right\}$ of $X$ such that $\overline{H(U,y)} \subset G(U,y)$ for each $U \in \mathcal{U}$ .

Now $\left\{V_y: y \in Y \right\}$ is an open cover of $Y$ . By the paracompactness of $Y$ , let $\left\{W_y: y \in Y \right\}$ be a locally finite open cover of $Y$ such that $\overline{W_y} \subset V_y$ for each $y \in Y$ . For each $U \in \mathcal{U}$ , define the following:

$W_U=\cup \left\{H(U,y) \times W_y: y \in Y \text{ such that } \overline{H(U,y) \times W_y} \subset U \right\}$

We claim that $\mathcal{W}=\left\{ W_U: U \in \mathcal{U} \right\}$ is a shrinking of $\mathcal{U}$ . First it is a cover of $X \times Y$ . Let $(x,t) \in X \times Y$ . Then $t \in W_y$ for some $y \in Y$ . There exists $U \in \mathcal{U}$ such that $x \in H(U,y)$ . Note the following.

$\overline{H(U,y) \times W_y} \subset \overline{H(U,y)} \times \overline{W_y} \subset G(U,y) \times V_y \subset U$

This means that $H(U,y) \times W_y \subset W_U$ . Since $(x,t) \in H(U,y) \times W_y$ , $(x,t) \in W_U$ . Thus $\mathcal{W}$ is an open cover of $X \times Y$ .

Now we show that $\mathcal{W}$ is a shrinking of $\mathcal{U}$ . Let $U \in \mathcal{U}$ . To show that $\overline{W_U} \subset U$ , let $(x,t) \in \overline{W_U}$ . Let $L$ be open in $Y$ such that $t \in L$ and that $L$ meets only finitely many $W_y$ , say for $y=y_1,y_2,\cdots,y_n$ . Immediately we have the following relations.

$\forall \ i=1,\cdots,n, \ \overline{W_{y_i}} \subset V_{y_i}$

$\forall \ i=1,\cdots,n, \ \overline{H(U,y_i)} \subset G(U,y_i)$

$\forall \ i=1,\cdots,n, \ \overline{H(U,y_i) \times W_{y_i}} \subset \overline{H(U,y_i)} \times \overline{W_{y_i}} \subset G(U,y_i) \times V_{y_i} \subset U$

Then it follows that

$\displaystyle (x,t) \in \overline{\bigcup \limits_{j=1}^n H(U,y_j) \times W_{y_j}}=\bigcup \limits_{j=1}^n \overline{H(U,y_j) \times W_{y_j}} \subset U$

Thus $U \in \mathcal{U}$ . This shows that $\mathcal{W}$ is a shrinking of $\mathcal{U}$ . $\square$

Remark
Theorem 2 is the Theorem 3.2 in [3]. Theorem 2 is a formulation of Theorem 3.2 [3] for the purpose of proving Theorem 3 below.

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Main Theorem

Theorem 3 (Main Theorem)
Let $X$ be a normal and countably compact space. Let $Y$ be a metric space. Then $X \times Y$ is a normal space.

Proof of Theorem 3
Let $\mathcal{U}$ be a 2-element open cover of $X \times Y$ . We show that $\mathcal{U}$ is shrinkable. This would mean that $X \times Y$ is normal (according to Theorem 1). To show that $\mathcal{U}$ is shrinkable, we show that the open cover $\mathcal{U}$ satisfies the two bullet points in Theorem 2.

Fix $y \in Y$ . Let $\left\{B_n: n=1,2,3,\cdots \right\}$ be a base at the point $y$ . Define $G_n$ as follows:

$G_n=\cup \left\{O \subset X: O \text{ is open such that } O \times B_n \subset U \text{ for some } U \in \mathcal{U} \right\}$

It is clear that $\mathcal{G}=\left\{G_n: n=1,2,3,\cdots \right\}$ is an open cover of $X$ . Since $X$ is countably compact, choose $m$ such that $\left\{G_1,G_2,\cdots,G_m \right\}$ is a cover of $X$ . Let $E_y=\bigcap_{j=1}^m B_j$ . We claim that $E_y$ is stable with respect to $\mathcal{U}$ . To see this, let $x \in X$ . Then $x \in G_j$ for some $j \le m$ . By the definition of $G_j$ , there is some open set $O_x \subset X$ such that $x \in O_x$ and $O_x \times B_j \subset U$ for some $U \in \mathcal{U}$ . Furthermore, $O_x \times E_y \subset O_x \times B_j \subset U$ .

To summarize: for each $y \in Y$ , there is an open set $E_y$ such that $y \in E_y$ and $E_y$ is stable with respect to the open cover $\mathcal{U}$ . Thus the first bullet point of Theorem 2 is satisfied. The open cover $\mathcal{U}$ is a 2-element open cover. Thus the second bullet point of Theorem 2 is satisfied. By Theorem 2, the open cover $\mathcal{U}$ is shrinkable. Thus $X \times Y$ is normal. $\square$

Corollary 4
Let $X$ be a normal and pseudocompact space. Let $Y$ be a metric space. Then $X \times Y$ is a normal space.

The corollary follows from the fact that any normal and pseudocompact space is countably compact (see here).

Remarks
The proof of Theorem 3 actually gives a more general result. Note that the second factor only needs to be paracompact and that every point has a countable base (i.e. first countable). The first factor $X$ has to be countably compact. The shrinking requirement for $X$ is flexible – if open covers of a certain size for $X$ are shrinkable, then open covers of that size for the product are shrinkable. We have the following corollaries.

Corollary 5
Let $X$ be a $\kappa$ -shrinking and countably compact space and let $Y$ be a paracompact first countable space. Then $X \times Y$ is a $\kappa$ -shrinking space.

Corollary 6
Let $X$ be a shrinking and countably compact space and let $Y$ be a paracompact first countable space. Then $X \times Y$ is a shrinking space.

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Remarks

The main theorem (Theorem 3) says that any normal and countably compact space is productively normal with one class of spaces, namely the metric spaces. Thus if one wishes to find a non-normal product space with one factor being countably compact, the other factor must not be a metric space. For example, if $W=\omega_1$ , the first uncountable ordinal with the ordered topology, then $W \times X$ is always normal for every metric $X$ . For non-normal example, $W \times C$ is not normal for any compact space $C$ with uncountable tightness (see Theorem 1 in this previous post). Another example, $W \times L_{\omega_1}$ is not normal where $L_{\omega_1}$ is the one-point Lindelofication of a discrete space of cardinality $\omega_1$ (follows from Example 1 and Theorem 7 in this previous post).

Another comment is that normal countably paracompact spaces are examples of Normal P-spaces. K. Morita defined the notion of P-space and he proved that a space $Y$ is a Normal P-space if and only if $X \times Y$ is normal for every metric space $X$ .

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Reference

Przymusinski T. C., Products of Normal Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 781-826, 1984.
Stone A. H., Paracompactness and Product Spaces, Bull. Amer. Math. Soc., Vol. 54, 977-982, 1948. (paper)
Yang L., The Normality in Products with a Countably Compact Factor, Canad. Math. Bull., Vol. 41 (2), 245-251, 1998. (abstract, paper)

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$\copyright \ 2017 \text{ by Dan Ma}$

kappa-Dowker space and the first conjecture of Morita

Posted on January 8, 2017 by Dan Ma

Recall the product space of the Michael line and the space of the irrational numbers. Even though the first factor is a normal space (in fact a paracompact space) and the second factor is a metric space, their product space is not normal. This is one of the classic examples demonstrating that normality is not well behaved with respect to product space. This post presents an even more striking result, i.e., for any non-discrete normal space $Y$ , there exists another normal space $X$ such that $X \times Y$ is not normal. The example of the non-normal product of the Michael line and the irrationals is not some isolated example. Rather it is part of a wide spread phenomenon. This result guarantees that no matter how nice a space $Y$ is, a counter part $X$ can always be found that the product of the two spaces is not normal. This result is known as Morita’s first conjecture and was proved by Atsuji and Rudin. The solution is based on a generalization of Dowker’s theorem and a construction done by Rudin. This post demonstrates how the solution is put together.

All spaces under consideration are Hausdorff.

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Morita’s First Conjecture

In 1976, K. Morita posed the following conjecture.

Morita’s First Conjecture

$Y$

$X \times Y$

$X$

$Y$

The proof given in this post is for proving the contrapositive of the above statement.

Morita’s First Conjecture

$Y$

$X$

$X \times Y$

Though the two forms are logically equivalent, the contrapositive form seems to have a bigger punch. The contrapositive form gives an association. Each non-discrete normal space is paired with a normal space to form a non-normal product. Examples of such pairings are readily available. Michael line is paired with the space of the irrational numbers (as discussed above). The Sogenfrey line is paired with itself. The first uncountable ordinal $\omega_1$ is paired with $\omega_1+1$ (see here) or paired with the cube $I^I$ where $I=[0,1]$ with the usual topology (see here). There are plenty of other individual examples that can be cited. In this post, we focus on a constructive proof of finding such a pairing.

Since the conjecture had been affirmed positively, it should no longer be called a conjecture. Calling it Morita’s first theorem is not appropriate since there are other results that are identified with Morita. In this discussion, we continue to call it a conjecture. Just know that it had been proven.

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Dowker’s Theorem

Next, we examine Dowker’s theorem, which characterizes normal countably paracompact spaces. The following is the statement.

Theorem 1 (Dowker’s Theorem)
Let $X$ be a normal space. The following conditions are equivalent.

The space $X$ is countably paracompact.
Every countable open cover of $X$ has a point-finite open refinement.
If $\left\{U_n: n=1,2,3,\cdots \right\}$ is an open cover of $X$ , there exists an open refinement $\left\{V_n: n=1,2,3,\cdots \right\}$ such that $\overline{V_n} \subset U_n$ for each $n$ .
The product space $X \times Y$ is normal for any compact metric space $Y$ .
The product space $X \times [0,1]$ is normal where $[0,1]$ is the closed unit interval with the usual Euclidean topology.
For each sequence $\left\{A_n \subset X: n=1,2,3,\cdots \right\}$ of closed subsets of $X$ such that $A_1 \supset A_2 \supset A_3 \supset \cdots$ and $\cap_n A_n=\varnothing$ , there exist open sets $B_1,B_2,B_3,\cdots$ such that $A_n \subset B_n$ for each $n$ such that $\cap_n B_n=\varnothing$ .

The theorem is discussed here and proved here. Any normal space that violates any one of the conditions in the theorem is said to be a Dowker space. One such space was constructed by Rudin in 1971 [2]. Any Dowker space would be one factor in a non-normal product space with the other factor being a compact metric space. Actually much more can be said.

The Dowker space constructed by Rudin is the solution of Morita’s conjecture for a large number of spaces. At minimum, the product of any infinite compact metric space and the Dowker space is not normal as indicated by Dowker’s theorem. Any nontrivial convergent sequence plus the limit point is a compact metric space since it is homeomorphic to $S=\left\{0 \right\} \cup \left\{\frac{1}{n}: n=1,2,3,\cdots \right\}$ (as a subspace of the real line). Thus Rudin’s Dowker space has non-normal product with $S$ . Furthermore, the product of Rudin’s Dowker space and any space containing a copy of $S$ is not normal.

Spaces that contain a copy of $S$ extend far beyond the compact metric spaces. Spaces that have lots of convergent sequences include first countable spaces, Frechet spaces and many sequential spaces (see here for an introduction for these spaces). Thus any Dowker space is an answer to Morita’s first conjecture for the non-discrete members of these classes of spaces. Actually, the range for the solution is wider than these spaces. It turns out that any space that has a countable non-discrete subspace would have a non-normal product with a Dowker space. These would include all the classes mentioned above (first countable, Frechet, sequential) as well as countably tight spaces and more.

Therefore, any Dowker space, a normal space that is not countably paracompact, is severely lacking in ability in forming normal product with another space. In order to obtain a complete solution to Morita’s first conjecture, we would need a generalized Dowker’s theorem.

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Shrinking Properties

The key is to come up with a generalized Dowker’s theorem, a theorem like Theorem 1 above, except that it is for arbitrary infinite cardinality. Then a $\kappa$ -Dowker space is a space that violates one condition in the theorem. That space would be a candidate for the solution of Morita’s first conjecture. Note that Theorem 1 is for the infinite countable cardinal $\omega$ only. Before stating the theorem, let’s gather all the notions that will go into the theorem.

Let $X$ be a space. Let $\mathcal{U}$ be an open cover of the space $X$ . The open cover $\mathcal{U}$ is said to be shrinkable if there is an open cover $\mathcal{V}=\left\{V(U): U \in \mathcal{U} \right\}$ such that $\overline{V(U)} \subset U$ for each $U \in \mathcal{U}$ . When this is the case, the open cover $\mathcal{V}$ is said to be a shrinking of $\mathcal{U}$ . If an open cover is shrinkable, we also say that the open cover can be shrunk (or has a shrinking).

Let $\kappa$ be a cardinal. The space $X$ is said to be a $\kappa$ -shrinking space if every open cover of cardinality $\le \kappa$ of the space $X$ is shinkable. The space $X$ is a shrinking space if it is a $\kappa$ -shrinking space for every cardinal $\kappa$ .

When a family of sets are indexed by ordinals, the notion of an increasing or decreasing family of sets is possible. For example, the family $\left\{A_\alpha \subset X: \alpha<\kappa \right\}$ of subsets of the space $X$ is said to be increasing if $A_\beta \subset A_\gamma$ whenever $\beta<\gamma$ . In other words, for an increasing family, the sets are getting larger whenever the index becomes larger. A decreasing family of sets is defined in the reverse way. These two notions are important for some shrinking properties discussed here – e.g. using an open cover that is increasing or using a family of closed sets that is decreasing.

In the previous discussion on shrinking spaces, two other shrinking properties are discussed – property $\mathcal{D}(\kappa)$ and property $\mathcal{B}(\kappa)$ . A space $X$ is said to have property $\mathcal{D}(\kappa)$ if every increasing open cover of cardinality $\le \kappa$ for the space $X$ is shrinkable. A space $X$ is said to have property $\mathcal{B}(\kappa)$ if every increasing open cover of cardinality $\le \kappa$ for the space $X$ has a shrinking that is increasing. See this previous post for a discussion on property $\mathcal{D}(\kappa)$ and property $\mathcal{B}(\kappa)$ .

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An Attempt for a Generalized Dowker’s Theorem

Let $\kappa$ be an infinite cardinal. The space $X$ is said to be a $\kappa$ -paracompact space if every open cover $\mathcal{U}$ of $X$ with $\lvert \mathcal{U} \lvert \le \kappa$ has a locally finite open refinement. Thus a space is paracompact if it is $\kappa$ -paracompact for every infinite cardinal $\kappa$ . Of course, an $\omega$ -paracompact space is a countably paracompact space.

For any infinite $\kappa$ , let $D_\kappa$ be a discrete space of size $\kappa$ . Let $p$ be a point not in $D_\kappa$ . Define the space $Y_\kappa=D_\kappa \cup \left\{p \right\}$ as follows. The subspace $D_\kappa$ is discrete as before. The open neighborhoods at $p$ are of the form $\left\{ p \right\} \cup B$ where $B \subset D_\kappa$ and $\lvert D_\kappa-B \lvert<\kappa$ . In other words, any open set containing $p$ contains all but less than $\kappa$ many discrete points.

Another concept that is needed is the cardinal function called minimal tightness. Let $Y$ be any space. Define the minimal tightness $mt(Y)$ as the least infinite cardinal $\kappa$ such that there is a non-discrete subspace of $Y$ of cardinality $\kappa$ . If $Y$ is a discrete space, then let $mt(Y)=0$ . For any non-discrete space $Y$ , $mt(Y)=\kappa$ for some infinite $\kappa$ . Note that for the space $Y_\kappa$ defined above would have $mt(Y_\kappa)=\kappa$ . For any space $Y$ , $mt(Y)=\omega$ if and only if $Y$ has a countable non-discrete subspace.

The following theorem can be called a $\kappa$ -Dowker’s Theorem.

Theorem 2
Let $X$ be a normal space. Let $\kappa$ be an infinite cardinal. Consider the following conditions.

The space $X$ is a $\kappa$ -paracompact space.
The space $X$ is a $\kappa$ -shrinking space.

For each open cover $\left\{U_\alpha: \alpha<\kappa \right\}$ of $X$ , there exists an open cover $\left\{V_\alpha: \alpha<\kappa \right\}$ such that $\overline{V_\alpha} \subset U_\alpha$ for each $\alpha<\kappa$ .

The space $X$ has property $\mathcal{D}(\kappa)$ .

For each increasing open cover $\left\{U_\alpha: \alpha<\kappa \right\}$ of $X$ , there exists an open cover $\left\{V_\alpha: \alpha<\kappa \right\}$ such that $\overline{V_\alpha} \subset U_\alpha$ for each $\alpha<\kappa$ .

For each decreasing family $\left\{F_\alpha: \alpha<\kappa \right\}$ of closed subsets of $X$ such that $\bigcap_{\alpha<\kappa} F_\alpha=\varnothing$ , there exists a family $\left\{G_\alpha: \alpha<\kappa \right\}$ of open subsets of $X$ such that $\bigcap_{\alpha<\kappa} G_\alpha=\varnothing$ and $F_\alpha \subset G_\alpha$ for each $\alpha<\kappa$ .
The space $X$ has property $\mathcal{B}(\kappa)$ .

For each increasing open cover $\left\{U_\alpha: \alpha<\kappa \right\}$ of $X$ , there exists an increasing open cover $\left\{V_\alpha: \alpha<\kappa \right\}$ such that $\overline{V_\alpha} \subset U_\alpha$ for each $\alpha<\kappa$ .

The product space $X \times Y_\kappa$ is a normal space.
The product space $X \times Y$ is a normal space for some space $Y$ with $mt(Y)=\kappa$ .

The following diagram shows how these conditions are related.

Diagram 1
$\displaystyle \begin{array}{ccccc} 1 &\text{ } & \Longrightarrow & \text{ } & 5 \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \Downarrow & \text{ } & \text{ } & \text{ } & \Updownarrow \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ 2 &\text{ } & \text{ } & \text{ } & 6 \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \Downarrow & \text{ } & \text{ } & \text{ } & \Downarrow \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ 3 & \text{ } & \Longleftarrow & \text{ } & 7 \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \Updownarrow & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ 4 & \text{ } & \text{ } & \text{ } & \text{ } \end{array}$

In addition to Diagram 1, we have the relations $5 \Longrightarrow 3$ and $2 \not \Longrightarrow 5$ .

Remarks
At first glance, Diagram 1 might give the impression that the conditions in the theorem form a loop. It turns out the strongest property is $\kappa$ -paracompactness (condition 1). Since condition 2 does not imply condition 5, condition 2 does not imply condition 1. Thus the conditions do not form a loop.

The implications $1 \Longrightarrow 2 \Longrightarrow 3 \Longleftarrow 5$ and $6 \Longrightarrow 7$ are immediate. The following implications are established in this previous post.

$3 \Longleftrightarrow 4$

$5 \Longleftrightarrow 6$ (Theorem 7)

$2 \not \Longrightarrow 5$ (Example 1)

The remaining implications to be shown are $1 \Longrightarrow 5$ and $7 \Longrightarrow 3$ .

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Proof of Theorem 2

$1 \Longrightarrow 5$
Let $\mathcal{U}=\left\{U_\alpha: \alpha<\kappa \right\}$ be an increasing open cover of $X$ . By $\kappa$ -paracompactness, let $\mathcal{G}$ be a locally finite open refinement of $\mathcal{U}$ . For each $\alpha<\kappa$ , define $W_\alpha$ as follows:

$W_\alpha=\cup \left\{G \in \mathcal{G}: G \subset U_\alpha \right\}$

Then $\mathcal{W}=\left\{W_\alpha: \alpha<\kappa \right\}$ is still a locally finite refinement of $\mathcal{U}$ . Since the space $X$ is normal, any locally finite open cover is shrinkable. Let $\mathcal{E}=\left\{E_\alpha: \alpha<\kappa \right\}$ be a shrinking of $\mathcal{W}$ . The open cover $\mathcal{E}$ is also locally finite. For each $\alpha$ , let $V_\alpha=\bigcup_{\beta<\alpha} E_\beta$ . Then $\mathcal{V}=\left\{V_\alpha: \alpha<\kappa \right\}$ is an increasing open cover of $X$ . Note that

$\overline{V_\alpha}=\overline{\bigcup_{\beta<\alpha} E_\beta}=\bigcup_{\beta<\alpha} \overline{E_\beta}$

since $\mathcal{E}$ is locally finite and thus closure preserving. Since $\mathcal{U}$ is increasing, $\overline{E_\beta} \subset W_\beta \subset U_\beta \subset U_\alpha$ for all $\beta<\alpha$ . This means that $\overline{V_\alpha} \subset U_\alpha$ for all $\alpha$ .

$7 \Longrightarrow 3$
Since condition 3 is equivalent to condition 4, we show $7 \Longrightarrow 4$ . Suppose that $X \times Y$ is normal where $Y$ is a space such that $mt(Y)=\kappa$ . Let $D=\left\{d_\alpha: \alpha<\kappa \right\}$ be a non-discrete subset of $Y$ . Let $p$ be a point such that $p \ne d_\alpha$ for all $\alpha$ and such that $p$ is a limit point of $D$ (this means that every open set containing $p$ contains some $d_\alpha$ ). Let $\mathcal{F}=\left\{F_\alpha: \alpha<\kappa \right\}$ be a decreasing family of closed subsets of $X$ such that $\bigcap_{\alpha<\kappa} F_\alpha=\varnothing$ . Define $H$ and $K$ as follows:

$H=\cup \left\{F_\alpha \times \left\{d_\alpha \right\}: \alpha<\kappa \right\}$

$K=X \times \left\{p \right\}$

The sets $H$ and $K$ are clearly disjoint. The set $K$ is clearly a closed subset of $X \times Y$ . To show that $H$ is closed, let $(x,y) \in (X \times Y)-H$ . Two cases to consider: $x \in F_0$ or $x \notin F_0$ where $F_0$ is the first closed set in the family $\mathcal{F}$ .

The first case $x \in F_0$ . Let $\beta<\kappa$ be least such that $x \notin F_\beta$ . Then $y \ne d_\gamma$ for all $\gamma<\beta$ since $(x,y) \in (X \times Y)-H$ . In the space $Y$ , any subset of cardinality $<\kappa$ is a closed set. Let $E=Y-\left\{d_\gamma: \gamma<\beta \right\}$ , which is open containing $y$ . Let $O \subset X$ be open such that $x \in O$ and $O \cap F_\beta=\varnothing$ . Then $(x,y) \in O \times E$ and $O \times E$ misses points of $H$ .

Now consider the second case $x \notin F_0$ . Let $O \subset X$ be open such that $x \in O$ and $O$ misses $F_0$ . Then $O \times Y$ is an open set containing $(x,y)$ such that $O \times Y$ misses $H$ . Thus $H$ is a closed subset of $X \times Y$ .

Since $X \times Y$ is normal, choose open $V \subset X \times Y$ such that $H \subset V$ and $\overline{V} \cap K=\varnothing$ . For each $\alpha<\kappa$ , define $G_\alpha$ as follows:

$G_\alpha=\left\{x \in X: (x,d_\alpha) \in V \right\}$

Note that each $G_\alpha$ is open in $X$ and that $F_\alpha \subset G_\alpha$ for each $\alpha<\kappa$ . We claim that $\bigcap_{\alpha<\kappa} G_\alpha=\varnothing$ . Let $x \in X$ . The point $(x,p)$ is in $K$ . Thus $(x,p) \notin \overline{V}$ . Choose an open set $L \times M$ such that $(x,p) \in L \times M$ and $(L \times M) \cap \overline{V}=\varnothing$ . Since $p \in M$ , there is some $\gamma<\kappa$ such that $d_\gamma \in M$ . Since $(x,d_\gamma) \notin \overline{V}$ , $(x,d_\gamma) \notin V$ . Thus $x \notin G_\gamma$ . This establishes the claim that $\bigcap_{\alpha<\kappa} G_\alpha=\varnothing$ .

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$\kappa$ -Dowker Space

Analogous to the Dowker space, a $\kappa$ -Dowker space is a normal space that violates one condition in Theorem 2. Since the seven conditions listed in Theorem 7 are not all equivalent, which condition to use? Condition 1 is the strongest condition since it implies all the other condition. At the lower left corner of Diagram 1 is condition 3, which follows from every other condition. Thus condition 3 (or 4) is the weakest property. An appropriate definition of a $\kappa$ -Dowker space is through negating condition 3 or condition 4. Thus, given an infinite cardinal $\kappa$ , a $\kappa$ -Dowker space is a normal space $X$ that satisfies the following condition:

$\left\{F_\alpha: \alpha<\kappa \right\}$

$X$

$\bigcap_{\alpha<\kappa} F_\alpha=\varnothing$

$\left\{G_\alpha: \alpha<\kappa \right\}$

$X$

$F_\alpha \subset G_\alpha$

$\alpha$

$\bigcap_{\alpha<\kappa} G_\alpha \ne \varnothing$

The definition of $\kappa$ -Dowker space is through negating condition 4. Of course, negating condition 3 would give an equivalent definition.

When $\kappa$ is the countably infinite cardinal $\omega$ , a $\kappa$ -Dowker space is simply the ordinary Dowker space constructed by M. E. Rudin [2]. Rudin generalized the construction of the ordinary Dowker space to obtain a $\kappa$ -Dowker space for every infinite cardinal $\kappa$ [4]. The space that Rudin constructed in [4] would be a normal space $X$ such that condition 4 of Theorem 2 is violated. This means that the space $X$ would violate condition 7 in Theorem 2. Thus $X \times Y$ is not normal for every space $Y$ with $mt(Y)=\kappa$ .

Here’s the solution of Morita’s first conjecture. Let $Y$ be a normal and non-discrete space. Determine the least cardinality $\kappa$ of a non-discrete subspace of $Y$ . Obtain the $\kappa$ -Dowker space $X$ as in [4]. Then $X \times Y$ is not normal according to the preceding paragraph.

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Remarks

Answering Morita’s first conjecture is a two-step approach. First, figure out what a generalized Dowker’s theorem should be. Then a $\kappa$ -Dowker space is one that violates an appropriate condition in the generalized Dowker’s theorem. By violating the right condition in the theorem, we have a way to obtain non-normal product space needed in the answer. The second step is of course the proof of the existence of a space that violates the condition in the generalized Dowker’s theorem.

Figuring out the form of the generalized Dowker’s theorem took some work. It is more than just changing the countable infinite cardinal in Dowker’s theorem (Theorem 1 above) to an arbitrary infinite cardinal. This is because the conditions in Theorem 1 are unequal when the cardinality is changed to an uncountable one.

We take the cue from Rudin’s chapter on Dowker spaces [3]. In the last page of that chapter, Rudin pointed out the conditions that should go into a generalized Dowker’s theorem. However, the explanation of the relationship among the conditions is not clear. The previous post and this post are an attempt to sort out the conditions and fill in as much details as possible.

Rudin’s chapter did have the right condition for defining $\kappa$ -Dowker space. It seems that prior to the writing of that chapter, there was some confusion on how to define a $\kappa$ -Dowker space, i.e. a condition in the theorem the violation of which would give a $\kappa$ -Dowker space. If the condition used is a stronger property, the violation may not yield enough information to get non-normal products. According to Diagram 1, condition 3 in Theorem 2 is the right one to use since it is the weakest condition and is down streamed from the conditions about normal product space. So the violation of condition 3 would answer Morita’s first conjecture.

We do not discuss the other step in the solution in any details. Any interested reader can review Rudin’s construction in [2] and [4]. The $\kappa$ -Dowker space is an appropriate subspace of a product space with the box topology.

One interesting observation about the ordinary Dowker space (the one that violates a condition in Theorem 1) is that the product of any Dowker space and any space with a countable non-discrete subspace is not normal. This shows that Dowker space is badly non-productive with respect to normality. This fact is actually not obvious in the usual formulation of Dowker’s theorem (Theorem 1 above). What makes this more obvious in the direction $7 \Longrightarrow 3$ in Theorem 2. For the countably infinite case, $7 \Longrightarrow 3$ is essentially this: If $X \times Y$ is normal where $Y$ has a countable non-discrete subspace, then $X$ is not a Dowker space. Thus if the goal is to find a non-normal product space, a Dowker space should be one space to check.

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Loose Ends

In the course of working on the contents in this post and the previous post, there are some questions that we do not know how to answer and have not spent time to verify one way or the other. Possibly there are some loose ends to tie. They for the most parts are not open questions, but they should be interesting questions to consider.

For the $\kappa$ -Dowker’s theorem (Theorem 2), one natural question is on the relative strengths of the conditions. It will be interesting to find out the implications not shown in Diagram 1. For example, for the three shrinking properties (conditions 2, 3 and 5), it is straightforward from definition that $2 \Longrightarrow 3$ and $5 \Longrightarrow 3$ . The example of $X=\omega_1$ (the first uncountable ordinal) shows that $2 \not \Longrightarrow 5$ and hence $2 \not \Longrightarrow 1$ . What about $3 \Longrightarrow 2$ ? In [5], Beslagic and Rudin showed that $3 \not \Longrightarrow 2$ using $\Diamond ^{++}$ . A natural question would be: can there be ZFC example? Perhaps searching on more recent papers can yield some answers.

Another question is $5 \Longrightarrow 1$ ? The answer is no with the example being a Navy space – Example 7.6 in p. 194 [1]. The other two directions that have not been accounted for are: $7 \Longrightarrow 6$ and $3 \Longrightarrow 7$ ? We do not know the answer.

Another small question that we come across is about $X=\omega_1$ (the first uncountable ordinal). This is an example for showing $2 \not \Longrightarrow 5$ . Thus condition 6 is false. Thus $X \times Y_{\omega_1}$ is not normal. Here $Y_{\omega_1}$ is simply the one-point Lindelofication of a discrete space of cardinality $\omega_1$ . The question is: is condition 7 true for $X=\omega_1$ ? The product of $X=\omega_1$ and $Y_{\omega_1}$ (a space with minimal tightness $\omega_1$ ) is not normal. Is there a normal $X \times Y$ where $Y$ is another space with minimal tightness $\omega_1$ ?

Dowker’s theorem and $\kappa$ -Dowker’s theorem show that finding a normal space that is not shrinking is not a simple matter. To find a normal space that is not countably shrinking took 20 years (1951 to 1971). For any uncountable $\kappa$ , the $\kappa$ -Dowker space that is based on the same construction of an ordinary Dowker space is also a space that is not $\kappa$ -shrinking. With an uncountable $\kappa$ , is the $\kappa$ -Dowker space countably shrinking? This is not obvious one way or the other just from the definition of $\kappa$ -Dowker space. Perhaps there is something obvious and we have not connected the dots. Perhaps we need to go into the definition of the $\kappa$ -Dowker space in [4] to show that it is countably shrinking. The motivation is that we tried to find a normal space that is countably shrinking but not $\kappa$ -shrinking for some uncountable $\kappa$ . It seems that the $\kappa$ -Dowker space in [4] is the natural candidate.

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Reference

Morita K., Nagata J.,Topics in General Topology, Elsevier Science Publishers, B. V., The Netherlands, 1989.
Rudin M. E., A Normal Space $X$ for which $X \times I$ is not Normal, Fund. Math., 73, 179-486, 1971. (link)
Rudin M. E., Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, (1984) 761-780.
Rudin M. E., $\kappa$ -Dowker Spaces, Czechoslovak Mathematical Journal, 28, No.2, 324-326, 1978. (link)
Rudin M. E., Beslagic A.,Set-Theoretic Constructions of Non-Shrinking Open Covers, Topology Appl., 20, 167-177, 1985. (link)
Yasui Y., On the Characterization of the $\mathcal{B}$ -Property by the Normality of Product Spaces, Topology and its Applications, 15, 323-326, 1983. (abstract and paper)
Yasui Y., Some Characterization of a $\mathcal{B}$ -Property, TSUKUBA J. MATH., 10, No. 2, 243-247, 1986.

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$\copyright \ 2017 \text{ by Dan Ma}$

Spaces with shrinking properties

Posted on January 5, 2017 by Dan Ma

Certain covering properties and separation properties allow open covers to shrink, e.g. paracompact spaces, normal spaces, and countably paracompact spaces. The shrinking property is also interesting on its own. This post gives a more in-depth discussion than the one in the previous post on countably paracompact spaces. After discussing shrinking spaces, we introduce three shrinking related properties. These properties show that there is a deep and delicate connection among shrinking properties and normality in products. This post is also a preparation for the next post on $\kappa$ -Dowker space and Morita’s first conjecture.

All spaces under consideration are Hausdorff and normal or Hausdorff and regular (if not normal).

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Shrinking Spaces

Let $X$ be a space. Let $\mathcal{U}$ be an open cover of $X$ . The open cover of $\mathcal{U}$ is said to be shrinkable if there is an open cover $\mathcal{V}=\left\{V(U): U \in \mathcal{U} \right\}$ of $X$ such that $\overline{V(U)} \subset U$ for each $U \in \mathcal{U}$ . When this is the case, the open cover $\mathcal{V}$ is said to be a shrinking of $\mathcal{U}$ . If an open cover is shrinkable, we also say that the open cover can be shrunk (or has a shrinking). Whenever an open cover has a shrinking, the shrinking is indexed by the open cover that is being shrunk. Thus if the original cover is indexed in a certain way, e.g. $\left\{U_\alpha: \alpha<\kappa \right\}$ , then a shrinking has the same indexing, e.g. $\left\{V_\alpha: \alpha<\kappa \right\}$ .

A space $X$ is a shrinking space if every open cover of $X$ is shrinkable. The property can also be broken up according to the cardinality of the open cover. Let $\kappa$ be a cardinal. A space $X$ is $\kappa$ -shrinking if every open cover of cardinality $\le \kappa$ for $X$ is shrinkable. A space $X$ is countably shrinking if it is $\omega$ -shrinking.

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Examples of Shrinking

Let’s look at a few situations where open covers can be shrunk either all the time or on a limited basis. For a normal space, certain covers can be shrunk as indicated by the following theorem.

Theorem 1
The following conditions are equivalent.

The space $X$ is normal.
Every point-finite open cover of $X$ is shrinkable.
Every locally finite open cover of $X$ is shrinkable.
Every finite open cover of $X$ is shrinkable.
Every two-element open cover of $X$ is shrinkable.

In a normal space, all finite open covers are shrinkable. In general, an infinite open cover of a normal space does not have to be shrinkable unless it is a point-finite or locally finite open cover.

The theorem of C. H. Dowker states that a normal space $X$ is countably paracompact if and only every countable open cover of $X$ is shrinkable if and only if the product space $X \times Y$ is normal for every compact metric space $Y$ if and only if the product space $X \times [0,1]$ is normal. The theorem is discussed here. A Dowker space is a normal space that violates the theorem. Thus any Dowker space has a countably infinite open cover that cannot be shrunk, or equivalently a normal space that forms a non-normal product with a compact metric space. Thus the notion of shrinking has a connection with normality in the product spaces. A Dowker space space was constructed by M. E. Rudin in ZFC [2]. So far Rudin’s example is essentially the only ZFC Dowker space. This goes to show that finding a normal space that is not countably shrinking is not a trivial matter.

Several facts can be derived easily from Theorem 1 and Dowker’s theorem. For clarity, they are called out as corollaries.

Corollary 2

All shrinking spaces are normal.
All shrinking spaces are normal and countably paracompact.
Any normal and metacompact space is a shrinking space.

For the first corollary, if every open cover of a space can be shrunk, then all finite open covers can be shrunk and thus the space must be normal. As indicated above, Dowker’s theorem states that in a normal space, countably paracompactness is equivalent to countably shrinking. Thus any shrinking space is normal and countably paracompact.

Though an infinite open cover of a normal space may not be shrinkable, adding an appropriate covering property to any normal space will make it into a shrinking space. An easy way is through point-finite open covers. If every open cover has a point-finite open refinement (i.e. a metacompact space), then the point-finite open refinement can be shrunk (if the space is also normal). Thus the third corollary is established. Note that the metacompact is not the best possible result. For example, it is known that any normal and submetacompact space is a shrinking space – see Theorem 6.2 of [1].

In paracompact spaces, all open covers can be shrunk. One way to see this is through Corollary 2. Any paracompact space is normal and metacompact. It is also informative to look at the following characterization of paracompact spaces.

Theorem 3
A space $X$ is paracompact if and only if every open cover $\left\{U_\alpha: \alpha<\kappa \right\}$ of $X$ has a locally finite open refinement $\left\{V_\alpha: \alpha<\kappa \right\}$ such that $\overline{V_\alpha} \subset U_\alpha$ for each $\alpha$ .

A proof can be found here. Thus every open cover of a paracompact space can be shrunk by a locally finite shrinking. To summarize, we have discussed the following implications.

Diagram 1

$\displaystyle \begin{aligned} \text{Paracompact} \Longrightarrow & \text{ Normal + Metacompact} \\&\ \ \ \ \ \ \Big \Downarrow \\&\text{ Shrinking} \\&\ \ \ \ \ \ \Big \Downarrow \\& \text{ Normal + Countably Paracompact} \\&\ \ \ \ \ \ \Big \Downarrow \\& \text{ Normal} \end{aligned}$

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Three Shrinking Related Properties

None of the implications in Diagram 1 can be reversed. The last implication in the diagram cannot be reversed due to Rudin’es Dowker space. One natural example to look for would be spaces that are normal and countably paracompact but fail in shrinking at some uncountable cardinal. As indicated by the the theorem of C. H, Dowker, the notion of shrinking is intimately connected to normality in product spaces $X \times Y$ . To further investigate, consider the following three properties.

Let $X$ be a space. Let $\kappa$ be an infinite cardinal. Consider the following three properties.

The space $X$ is $\kappa$ -shrinking if and only if any open cover of cardinality $\le \kappa$ for the space $X$ is shrinkable, i.e. the following condition holds.

$\left\{U_\alpha: \alpha<\kappa \right\}$

$X$

$\left\{V_\alpha: \alpha<\kappa \right\}$

$\overline{V_\alpha} \subset U_\alpha$

$\alpha<\kappa$

The space $X$ has Property $\mathcal{D}(\kappa)$ if and only if every increasing open cover of cardinality $\le \kappa$ for the space $X$ is shrinkable, i.e. the following holds.

$\left\{U_\alpha: \alpha<\kappa \right\}$

$X$

$\left\{V_\alpha: \alpha<\kappa \right\}$

$\overline{V_\alpha} \subset U_\alpha$

$\alpha<\kappa$

The space $X$ has Property $\mathcal{B}(\kappa)$ if and only if the following holds.

$\left\{U_\alpha: \alpha<\kappa \right\}$

$X$

$\left\{V_\alpha: \alpha<\kappa \right\}$

$\overline{V_\alpha} \subset U_\alpha$

$\alpha<\kappa$

A family $\left\{A_\alpha: \alpha<\kappa \right\}$ is increasing if $A_\alpha \subset A_\beta$ for any $\alpha<\beta<\kappa$ . It is decreasing if $A_\beta \subset A_\alpha$ for any $\alpha<\beta<\kappa$ .

In general, any space that is $\kappa$ -shrinking for all cardinals $\kappa$ is a shrinking space as defined earlier. Any space that has property $\mathcal{D}(\kappa)$ for all cardinals $\kappa$ is said to have property $\mathcal{D}$ . Any space that has property $\mathcal{B}(\kappa)$ for all cardinals $\kappa$ is said to have property $\mathcal{B}$ .

The first property $\kappa$ -shrinking is simply the shrinking property for open covers of cardinality $\le \kappa$ . The property $\mathcal{D}(\kappa)$ is $\kappa$ -shrinking with the additional requirement that the open covers to be shrunk must be increasing. It is clear that $\kappa$ -shrinking implies property $\mathcal{D}(\kappa)$ . The property $\mathcal{B}(\kappa)$ appears to be similar to $\mathcal{D}(\kappa)$ except that $\mathcal{B}(\kappa)$ has the additional requirement that the shrinking is also increasing. As a result $\mathcal{B}(\kappa)$ implies $\mathcal{D}(\kappa)$ . The following diagram shows the implications.

Diagram 2

$\displaystyle \begin{array}{ccccc} \kappa \text{-Shrinking} &\text{ } & \not \longrightarrow & \text{ } & \text{Property } \mathcal{B}(\kappa) \\ \text{ } & \searrow & \text{ } & \swarrow & \text{ } \\ \text{ } &\text{ } & \text{Property } \mathcal{D}(\kappa) & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \end{array}$

The implications in Diagram 2 are immediate. An example is given below showing that $\omega_1$ -shrinking does not imply property $\mathcal{B}(\omega_1)$ . If $\kappa=\omega$ , then all three properties are equivalent in normal spaces, as displayed in the following diagram. The proof is in Theorem 5.

Diagram 3

$\displaystyle \begin{array}{ccccc} \omega \text{-Shrinking} &\text{ } & \longrightarrow & \text{ } & \text{Property } \mathcal{B}(\omega) \\ \text{ } & \nwarrow & \text{ } & \swarrow & \text{ } \\ \text{ } &\text{ } & \text{Property } \mathcal{D}(\omega) & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \end{array}$

The property $\mathcal{D}(\kappa)$ has a dual statement in terms of decreasing closed sets. The following theorem gives the dual statement.

Theorem 4
Let $X$ be a normal space. Let $\kappa$ be an infinite cardinal. The following two properties are equivalent.

The space $X$ has property $\mathcal{D}(\kappa)$ .
For each decreasing family $\left\{F_\alpha: \alpha<\kappa \right\}$ of closed subsets of $X$ such that $\bigcap_{\alpha<\kappa} F_\alpha=\varnothing$ , there exists a family $\left\{G_\alpha: \alpha<\kappa \right\}$ of open subsets of $X$ such that $\bigcap_{\alpha<\kappa} G_\alpha=\varnothing$ and $F_\alpha \subset G_\alpha$ for each $\alpha<\kappa$ .

First bullet implies second bullet
Let $\left\{F_\alpha: \alpha<\kappa \right\}$ be a decreasing family of closed subsets of $X$ with empty intersection. Then $\left\{U_\alpha: \alpha<\kappa \right\}$ is an increasing family of open subsets of $X$ where $U_\alpha=X-F_\alpha$ . Let $\left\{V_\alpha: \alpha<\kappa \right\}$ be an open cover of $X$ such that $\overline{V_\alpha} \subset U_\alpha$ for each $\alpha$ . Then $\left\{G_\alpha: \alpha<\kappa \right\}$ where $G_\alpha=X-\overline{V_\alpha}$ is the needed open expansion.

Second bullet implies first bullet
Let $\left\{U_\alpha: \alpha<\kappa \right\}$ be an increasing open cover of $X$ . Then $\left\{F_\alpha: \alpha<\kappa \right\}$ is a decreasing family of closed subsets of $X$ where $F_\alpha=X-U_\alpha$ . Note that $\bigcap_{\alpha<\kappa} F_\alpha=\varnothing$ . Let $\left\{G_\alpha: \alpha<\kappa \right\}$ be a family of open subsets of $X$ such that $\bigcap_{\alpha<\kappa} G_\alpha=\varnothing$ and $F_\alpha \subset G_\alpha$ for each $\alpha$ . For each $\alpha$ , there is open set $W_\alpha$ such that $F_\alpha \subset W_\alpha \subset \overline{W_\alpha} \subset G_\alpha$ since $X$ is normal. For each $\alpha$ , let $V_\alpha=X-\overline{W_\alpha}$ . Then $\left\{V_\alpha: \alpha<\kappa \right\}$ is a family of open subsets of $X$ required by the first bullet. It is a cover because $\bigcap_{\alpha<\kappa} \overline{W_\alpha}=\varnothing$ . To show $\overline{V_\alpha} \subset U_\alpha$ , let $x \in \overline{V_\alpha}$ such that $x \notin U_\alpha$ . Then $x \in W_\alpha$ . Since $x \in \overline{V_\alpha}$ and $W_\alpha$ is open, $W_\alpha \cap V_\alpha \ne \varnothing$ . Let $y \in W_\alpha \cap V_\alpha$ . Since $y \in V_\alpha$ , $y \notin \overline{W_\alpha}$ , which means $y \notin W_\alpha$ , a contradiction. Thus $\overline{V_\alpha} \subset U_\alpha$ .

Now we show that the three properties in Diagram 3 are equivalent.

Theorem 5
Let $X$ be a normal space. Then the following implications hold.
$\omega$ -shrinking $\Longrightarrow$ Property $\mathcal{B}(\omega)$ $\Longrightarrow$ Property $\mathcal{D}(\omega)$ $\Longrightarrow$ $\omega$ -shrinking

Proof of Theorem 5
$\omega$ -shrinking $\Longrightarrow$ Property $\mathcal{B}(\omega)$
Suppose that $X$ is $\omega$ -shrinking. By Dowker’s theorem, $X \times (\omega+1)$ is a normal space. We can think of $\omega+1$ as a convergent sequence with $\omega$ as the limit point. Let $\left\{U_n:n=0,1,2,\cdots \right\}$ be an increasing open cover of $X$ . Define $H$ and $K$ as follows:

$H=\cup \left\{(X-U_n) \times \left\{n \right\}: n=0,1,2,\cdots \right\}$

$K=X \times \left\{\omega \right\}$

It is straightforward to verify that $H$ and $K$ are disjoint closed subsets of $X \times (\omega+1)$ . By normality, let $V$ and $W$ be disjoint open subsets of $X \times (\omega+1)$ such that $H \subset W$ and $K \subset V$ . For each integer $n=0,1,2,\cdots$ , define $V_n$ as follows:

$V_n=\left\{x \in X: \exists \ \text{open } O \subset X \text{ such that } x \in O \text{ and } O \times [n, \omega] \subset V \right\}$

The set $[n, \omega]$ consists of all integers $\ge n$ and the limit point $\omega$ . From the way the sets $V_n$ are defined, $\left\{V_n:n=0,1,2,\cdots \right\}$ is an increasing open cover of $X$ . The remaining thing to show is that $\overline{V_n} \subset U_n$ for each $n$ . Suppose that $x \in \overline{V_n}$ and $x \notin U_n$ . Then $(x,n) \in H$ by definition of $H$ . There exists an open set $E \times \left\{n \right\}$ such that $(x,n) \in E \times \left\{n \right\}$ and $(E \times \left\{n \right\}) \cap V=\varnothing$ . Since $E$ is an open set containing $x$ , $E \cap V_n \ne \varnothing$ . Let $y \in E \cap V_n$ . By definition of $V_n$ , there is some open set $O$ such that $y \in O$ and $O \times [n, \omega] \subset V$ , a contradiction since $(E \cap O) \times \left\{n \right\}$ is supposed to miss $V$ . Thus $\overline{V_n} \subset U_n$ for all integers $n$ .

The direction Property $\mathcal{B}(\omega)$ $\Longrightarrow$ Property $\mathcal{D}(\omega)$ is immediate.

Property $\mathcal{D}(\omega)$ $\Longrightarrow$ $\omega$ -shrinking
Consider the dual condition of $\mathcal{D}(\omega)$ in Theorem 4, which is equivalent to $\omega$ -shrinking according to Dowker’s theorem. $\square$

Remarks
The direction $\omega$ -shrinking $\Longrightarrow$ Property $\mathcal{B}(\omega)$ is true because $\omega$ -shrinking is equivalent to the normality in the product $X \times (\omega+1)$ . The same is not true when $\kappa$ becomes an uncountable cardinal. We now show that $\kappa$ -shrinking does not imply $\mathcal{B}(\kappa)$ in general.

Example 1
The space $X=\omega_1$ is the set of all ordinals less than $\omega_1$ with the ordered topology. Since it is a linearly ordered space, it is a shrinking space. Thus in particular it is $\omega_1$ -shrinking. To show that $X$ does not have property $\mathcal{B}(\omega_1)$ , consider the increasing open cover $\left\{U_\alpha: \alpha<\omega_1 \right\}$ where $U_\alpha=[0,\alpha)$ for each $\alpha<\omega_1$ . Here $[0,\alpha)$ consists of all ordinals less than $\alpha$ . Suppose $X$ has property $\mathcal{B}(\omega_1)$ . Then let $\left\{V_\alpha: \alpha<\omega_1 \right\}$ be an increasing open cover of $X$ such that $\overline{V_\alpha} \subset U_\alpha$ for each $\alpha$ .

Let $L$ be the set of all limit ordinals in $X$ . For each $\alpha \in L$ , $\alpha \notin U_\alpha$ and thus $\alpha \notin \overline{V_\alpha}$ . Thus there exists a countable ordinal $f(\alpha)<\alpha$ such that $(f(\alpha),\alpha]$ misses points in $\overline{V_\alpha}$ . Thus the map $f: L \rightarrow \omega_1$ is a pressing down map. By the pressing down lemma, there exists some $\alpha<\omega_1$ such that $S=f^{-1}(\alpha)$ is a stationary set in $\omega_1$ , which means that $S$ intersects with every closed and unbounded subset of $X=\omega_1$ . This means that for each $\gamma>\alpha$ , $(\alpha, \gamma]$ would miss $\overline{V_\gamma}$ . This means that for each $\gamma>\alpha$ , $\overline{V_\gamma} \subset [0,\alpha]$ . As a result $\left\{V_\alpha: \alpha<\omega_1 \right\}$ would not be a cover of $X$ , a contradiction. So $X$ does not have property $\mathcal{B}(\omega_1)$ . $\square$

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Property $\mathcal{B}(\kappa)$

Of the three properties discussed in the above section, we would like to single out property $\mathcal{B}(\kappa)$ . This property has a connection with normality in the product $X \times Y$ (see Theorem 7). First, we prove a lemma that is used in proving Theorem 7.

Lemma 6
Show that the property $\mathcal{B}(\kappa)$ is hereditary with respect to closed subsets.

Proof of Lemma 6
Let $X$ be a space with property $\mathcal{B}(\kappa)$ . Let $A$ be a closed subspace of $X$ . Let $\left\{U_\alpha \subset A: \alpha<\kappa \right\}$ be an increasing open cover of $A$ . For each $\alpha$ , let $W_\alpha$ be an open subset of $X$ such that $U_\alpha=W_\alpha \cap A$ . Since the open sets $U_\alpha$ are increasing, the open sets $W_\alpha$ can be chosen inductively such that $W_\alpha \supset W_\gamma$ for all $\gamma<\alpha$ . This will ensure that $W_\alpha$ will form an increasing cover.

Then $\left\{W_\alpha^* \subset X: \alpha<\kappa \right\}$ is an increasing open cover of $X$ where $W_\alpha^*=W_\alpha \cup (X-A)$ . By property $\mathcal{B}(\kappa)$ , let $\left\{E_\alpha \subset X: \alpha<\kappa \right\}$ be an increasing open cover of $X$ such that $\overline{E_\alpha} \subset W_\alpha^*$ . For each $\alpha$ , let $V_\alpha=E_\alpha \cap A$ . It can be readily verified that $\left\{V_\alpha \subset A: \alpha<\kappa \right\}$ is an increasing open cover of $A$ . Furthermore, $\overline{V_\alpha} \subset U_\alpha$ for each $\alpha$ (closure taken in $A$ ). $\square$

Let $\kappa$ be an infinite cardinal. Let $D_\kappa=\left\{d_\alpha: \alpha<\kappa \right\}$ be a discrete space of cardinality $\kappa$ . Let $p$ be a point not in $D_\kappa$ . Let $Y_\kappa=D_\kappa \cup \left\{p \right\}$ . Define a topology on $Y_\kappa$ by letting $D_\kappa$ be discrete and by letting open neighborhood of $p$ be of the form $\left\{p \right\} \cup E$ where $E \subset D_\kappa$ and $D_\kappa-E$ has cardinality less than $\kappa$ . Note the similarity between $Y_\kappa$ and the convergent sequence $\omega+1$ in the proof of Theorem 5.

Theorem 7
Let $X$ be a normal space. Then the product space $X \times Y_\kappa$ is normal if and only if $X$ has property $\mathcal{B}(\kappa)$ .

Remarks
The property $\mathcal{B}(\kappa)$ involves the shrinking of any increasing open cover with the added property that the shrinking is also increasing. The increasing shrinking is just what is needed to show that disjoint closed subsets of the product space can be separated.

Notations
Let’s set some notations that are useful in proving Theorem 7.

The set $[d_\alpha,p]$ is an open set in $Y_\kappa$ containing the point $p$ and is defined as follows.

$[d_\alpha,p]=\left\{d_\beta: \alpha \le \beta<\kappa \right\} \cup \left\{p \right\}$ .

For any two disjoint closed subsets $H$ and $K$ of the product space $X \times Y_\kappa$ , define the following sets.

For each $\alpha<\kappa$ , let $H_\alpha=H \cap (X \times \left\{d_\alpha \right\})$ and $K_\alpha=K \cap (X \times \left\{d_\alpha \right\})$ .
Let $H_p=H \cap (X \times \left\{p \right\})$ and $K_p=K \cap (X \times \left\{p \right\})$ .
For each $\alpha<\kappa$ , choose open $O_\alpha \subset X$ such that $G_\alpha=O_\alpha \times \left\{d_\alpha \right\}$ , $H_\alpha \subset G_\alpha$ and $\overline{G_\alpha} \cap K_\alpha=\varnothing$ (due to normality of $X$ ).
Choose open $O_p \subset X$ such that $G_p=O_p \times \left\{p \right\}$ , $H_p \subset G_p$ and $\overline{G_p} \cap K_p=\varnothing$ (due to normality of $X$ ).

Proof of Theorem 7
Suppose that $X$ has property $\mathcal{B}(\kappa)$ . Let $H$ and $K$ be two disjoint closed sets of $X \times Y_\kappa$ . Consider the following cases based on the locations of the closed sets $H$ and $K$ .

$H \subset X \times D_\kappa$

$K \subset X \times D_\kappa$

$H=X \times \left\{p\right\}$

$H$

$K$

$X \times \left\{p\right\}$

$H$

$K$

$X \times \left\{p\right\}$

Remarks
Case 1 is easy. Case 2a is the pivotal case. Case 2b and Case 3 use a similar idea. The result in Theorem 7 is found in [1] (Theorem 6.9 in p. 189) and [4]. The authors in these two sources claimed that Case 2a is the only case that matters, citing a lemma in another source. The lemma was not stated in these two sources and the source for the lemma is a PhD dissertation that is not readily available. Case 3 essentially uses the same idea but it has enough differences. For the sake of completeness, we work out all the cases. Case 3 applies property $\mathcal{B}(\kappa)$ twice. Despite the complicated notations, the essential idea is quite simple. If any reader finds the proof too long, just understand Case 2a and then get the gist of how the idea is applied in Case 2b and Case 3.

Case 1.
$H \subset X \times D_\kappa$ and $K \subset X \times D_\kappa$ .

Let $M =\bigcup_{\alpha<\kappa} G_\alpha$ . It is clear that $H \subset M$ and $\overline{M} \cap K=\varnothing$ .

Case 2a.
Assume that $H=X \times \left\{p\right\}$ . We now proceed to separate $H$ and $K$ with disjoint open sets. For each $\alpha<\kappa$ , define $U_\alpha$ as follows:

$U_\alpha=\cup \left\{O \subset X: O \text{ is open such that } (O \times [d_\alpha,p]) \cap K =\varnothing \right\}$

Then $\left\{U_\alpha: \alpha<\kappa \right\}$ is an increasing open cover of $X$ . By property $\mathcal{B}(\kappa)$ , there is an increasing open cover $\mathcal{V}=\left\{V_\alpha: \alpha<\kappa \right\}$ of $X$ such that $\overline{V_\alpha} \subset U_\alpha$ for each $\alpha$ . The shrinking $\mathcal{V}$ allows us to define an open set $G$ such that $H \subset G$ and $\overline{G} \cap K=\varnothing$ .

Let $G=\cup \left\{V_\alpha \times [d_\alpha,p]: \alpha<\kappa \right\}$ . It is clear that $H \subset G$ . Next, we show that $\overline{G} \cap K=\varnothing$ . Suppose that $(x,d_\alpha) \in K$ . Then $(x,d_\alpha) \notin U_\alpha \times [d_\alpha,p]$ . As a result, $(x,d_\alpha) \notin \overline{V_\alpha} \times [d_\alpha,p]$ . Let $O \subset X$ be open such that $x \in O$ and $(O \times \left\{d_\alpha \right\}) \cap (\overline{V_\alpha} \times [d_\alpha,p])=\varnothing$ . Since $V_\beta \subset V_\alpha$ for all $\beta<\alpha$ , it follows that $(O \times \left\{d_\alpha \right\}) \cap (V_\beta \times [d_\beta,p])=\varnothing$ for all $\beta < \alpha$ . It is clear that $(O \times \left\{d_\alpha \right\}) \cap (V_\gamma \times [d_\gamma,p])=\varnothing$ for all $\gamma>\alpha$ . What has been shown is that there is an open set containing the point $(x,d_\alpha)$ that contains no point of $G$ . This means that $(x,d_\alpha) \notin \overline{G}$ . We have established that $\overline{G} \cap K=\varnothing$ .

Case 2b.
Exactly one of $H$ and $K$ intersect the set $X \times \left\{p\right\}$ . We assume that $H$ is the set that intersects the set $X \times \left\{p\right\}$ . The only difference between Case 2b and Case 2a is that there can be points of $H$ outside of $X \times \left\{p\right\}$ in Case 2b.

Now proceed as in Case 2a. Obtain the open cover $\left\{U_\alpha: \alpha<\kappa \right\}$ , the open cover $\left\{V_\alpha: \alpha<\kappa \right\}$ and the open set $G$ as in Case 2a. Let $M=G \cup (\bigcup_{\alpha<\kappa} G_\alpha)$ . It is clear that $H \subset M$ . We claim that $\overline{M} \cap K=\varnothing$ . Suppose that $(x,d_\gamma) \in K$ . Since $\overline{G} \cap K=\varnothing$ (as in Case 2a), there exists open set $W=O \times \left\{ d_\gamma \right\}$ such that $(x,d_\gamma) \in W$ and $W \cap \overline{G}=\varnothing$ . There also exists open $W_1 \subset W$ such that $(x,d_\gamma) \in W_1$ and $W_1 \cap \overline{G_\gamma}=\varnothing$ . It is clear that $W_1 \cap G_\beta=\varnothing$ for all $\beta \ne \gamma$ . This means that $W_1$ is an open set containing the point $(x,d_\gamma)$ such that $W_1$ misses the open set $M$ . Thus $\overline{M} \cap K=\varnothing$ .

Case 3.
Both $H$ and $K$ intersect the set $X \times \left\{p\right\}$ .

Now project $H_p$ and $K_p$ onto the space $X$ .

$H_p^*=\left\{x \in X: (x,p) \in H_p \right\}$

$K_p^*=\left\{x \in X: (x,p) \in K_p \right\}$

Note that $H_p^*$ is simply the copy of $H_p$ and $K_p^*$ is the copy of $K_p$ in $X$ . Since $X$ is normal, choose disjoint open sets $E_1$ and $E_1$ such that $H_p^* \subset E_1$ and $K_p^* \subset E_2$ .

Let $A_1=\overline{E_1}$ and $B_1=X-K_p^*$ . Let $A_2=\overline{E_2}$ and $B_2=X-H_p^*$ . Note that $A_1$ is closed in $X$ , $B_1$ is open in $X$ and $A_1 \subset B_1$ . Similarly $A_2$ is closed in $X$ , $B_2$ is open in $X$ and $A_2 \subset B_2$ .

We now define two increasing open covers using property $\mathcal{B}(\kappa)$ . Define $U_{\alpha,1}$ and $T_{\alpha,1}$ and $U_{\alpha,2}$ and $T_{\alpha,2}$ as follows:

$U_{\alpha,1}=\cup \left\{O \subset B_1: O \text{ is open such that } (O \times [d_\alpha,p]) \cap K =\varnothing \right\}$

$T_{\alpha,1}=U_{\alpha,1} \cap A_1$

$U_{\alpha,2}=\cup \left\{O \subset B_2: O \text{ is open such that } (O \times [d_\alpha,p]) \cap H =\varnothing \right\}$

$T_{\alpha,2}=U_{\alpha,2} \cap A_2$

The open cover $\mathcal{T}_1=\left\{T_{\alpha,1}: \alpha<\kappa \right\}$ is an increasing open cover of $A_1$ . The open cover $\mathcal{T}_2=\left\{T_{\alpha,2}: \alpha<\kappa \right\}$ is an increasing open cover of $A_2$ .By property $\mathcal{B}(\kappa)$ of $A_1$ and $A_2$ , both covers have the following as shrinking (by Lemma 6). The two shrinkings are:

$\mathcal{V}_1=\left\{V_{\alpha,1} \subset A_1: \alpha<\kappa \right\}$

$\mathcal{V}_2=\left\{V_{\alpha,2} \subset A_2: \alpha<\kappa \right\}$

such that

$\overline{V_{\alpha,1}} \subset T_{\alpha,1}$

$\overline{V_{\alpha,2}} \subset T_{\alpha,2}$

for each $\alpha<\kappa$ and such that both $\mathcal{V}_1$ and $\mathcal{V}_2$ are increasing open covers. Note that the closure $\overline{V_{\alpha,1}}$ is taken in $A_1$ and the closure $\overline{V_{\alpha,2}}$ is taken in $A_2$ .

For each $\alpha$ , let $W_{\alpha,1}$ be the interior of $V_{\alpha,1}$ and $W_{\alpha,2}$ be the interior of $V_{\alpha,2}$ (with respect to $X$ ). Note that $W_{\alpha,1}$ is meaningful since $V_{\alpha,1}$ is a subset of the closure of the open set $E_1$ . Similar observation for $W_{\alpha,2}$ . To make the rest of the argument easier to see, note the following fact about $W_{\alpha,1}$ and $W_{\alpha,2}$ .

$\overline{W_{\alpha,1}} \subset \overline{V_{\alpha,1}} \subset T_{\alpha,1} \subset U_{\alpha,1}$

$X$

$\overline{W_{\alpha,2}} \subset \overline{V_{\alpha,2}} \subset T_{\alpha,2} \subset U_{\alpha,2}$ (closure with respect to $X$ )

For each $\alpha<\kappa$ , choose open set $O_\alpha \subset X$ such that

$L_\alpha=O_\alpha \times \left\{d_\alpha \right\}$

$H_\alpha \subset L_\alpha$

$\overline{L_\alpha} \cap K_\alpha=\varnothing$

$L_\alpha \cap (\overline{W_{\alpha,2}} \times [d_\alpha,p])=\varnothing$

The last point is possible because $U_{\alpha,2} \times [d_\alpha,p]$ misses $H$ and $\overline{W_{\alpha,2}} \subset U_{\alpha,2}$ . Define the open sets $G$ and $M$ as follows:

$G=\cup \left\{W_{\alpha,1} \times [d_\alpha,p]: \alpha<\kappa \right\}$

$M=G \cup (\bigcup_{\alpha<\kappa} L_\alpha)$

It is clear that $H \subset M$ . We claim that $\overline{M} \cap K=\varnothing$ . To this end, we show that if $(x,y) \in K$ , then $(x,y) \notin \overline{M}$ . If $(x,y) \in K$ , then either $(x,y)=(x,d_\gamma)$ for some $\gamma$ or $(x,y)=(x,p)$ .

Let $(x,d_\gamma) \in K$ . Note that $(x,d_\gamma) \notin U_{\gamma,1} \times [d_\gamma,p]$ . Since $\overline{W_{\gamma,1}} \subset \overline{V_{\gamma,1}} \subset T_{\gamma,1} \subset U_{\gamma,1}$ , $(x,d_\gamma) \notin \overline{W_{\gamma,1}} \times [d_\gamma,p]$ . Choose an open set $O \subset X$ such that $x \in O$ and $C=O \times \left\{d_\gamma \right\}$ misses $\overline{W_{\gamma,1}} \times [d_\gamma,p]$ . Note that $C$ misses $W_{\beta,1} \times [d_\beta,p]$ for all $\beta<\gamma$ since $W_{\beta,1} \subset W_{\gamma,1}$ for all $\beta<\gamma$ . It is clear that $C$ misses $W_{\beta,1} \times [d_\beta,p]$ for all $\beta>\gamma$ .

We can also choose open $C_1 \subset C$ such that $(x,d_\gamma) \in C_1$ and $C_1$ misses $\overline{L_\gamma}$ . It is clear that $C_1$ misses $L_\beta$ for all $\beta \ne \gamma$ . Thus there is an open set $C_1$ containing the point $(x,d_\gamma)$ such that $C_1$ contains no point of $M$ .

Let $(x,p) \in K$ . First we find an open set $Q$ containing $(x,p)$ such that $Q$ misses $G$ . From the way the open sets $U_{\alpha,1}$ are defined, it follows that $(x,p) \notin \overline{W_{\alpha,1}} \times [d_\alpha,p]$ for all $\alpha$ . Furthermore $W_{\alpha,1} \subset \overline{A_1}$ . Thus $Q=(X-\overline{A_1}) \times Y_\kappa$ is the desired open set. On the other hand, there exists $\alpha<\kappa$ such that $x \in W_{\alpha,2}$ . Note that $L_\gamma$ are chosen so that $(W_{\gamma,2} \times [d_\gamma,p]) \cap L_\gamma=\varnothing$ for all $\gamma$ . Since $W_{\alpha,2} \subset W_{\beta,2}$ for all $\beta \ge \alpha$ , $(W_{\alpha,2} \times [d_\alpha,p]) \cap L_\beta=\varnothing$ for all $\beta \ge \alpha$ . Thus the open set $W_{\alpha,2} \times [d_\alpha,p]$ contains no points of $L_\gamma$ for any $\gamma$ . Then the open set $Q \cap (W_{\alpha,2} \times [d_\alpha,p])$ contains no point of $M$ . This means that $(x,p) \notin \overline{M}$ . Thus $\overline{M} \cap K=\varnothing$ .

In each of the four cases (1, 2a, 2b and 3), there exists an open set $M \subset X \times Y_\kappa$ such that $H \subset M$ and $\overline{M} \cap K=\varnothing$ . This completes the proof that $X \times Y_\kappa$ is normal assuming that $X$ has property $\mathcal{B}(\kappa)$ .

Now the other direction. Suppose that $X \times Y_\kappa$ is normal. Then it can be shown that $X$ has property $\mathcal{B}(\kappa)$ . The proof is similar to the proof for $\omega$ -shrinking $\Longrightarrow$ Property $\mathcal{B}(\omega)$ in Theorem 5. $\square$

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Reference

Morita K., Nagata J.,Topics in General Topology, Elsevier Science Publishers, B. V., The Netherlands, 1989.
Rudin M. E., A Normal Space $X$ for which $X \times I$ is not Normal, Fund. Math., 73, 179-486, 1971. (link)
Rudin M. E., Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, (1984) 761-780.
Yasui Y., On the Characterization of the $\mathcal{B}$ -Property by the Normality of Product Spaces, Topology and its Applications, 15, 323-326, 1983. (abstract and paper)
Yasui Y., Some Characterization of a $\mathcal{B}$ -Property, TSUKUBA J. MATH., 10, No. 2, 243-247, 1986.

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$\copyright \ 2017 \text{ by Dan Ma}$

Countably paracompact spaces

Posted on December 8, 2016 by Dan Ma

This post is a basic discussion on countably paracompact space. A space is a paracompact space if every open cover has a locally finite open refinement. The definition can be tweaked by saying that only open covers of size not more than a certain cardinal number $\tau$ can have a locally finite open refinement (any space with this property is called a $\tau$ -paracompact space). The focus here is that the open covers of interest are countable in size. Specifically, a space is a countably paracompact space if every countable open cover has a locally finite open refinement. Even though the property appears to be weaker than paracompact spaces, the notion of countably paracompactness is important in general topology. This post discusses basic properties of such spaces. All spaces under consideration are Hausdorff.

Basic discussion of paracompact spaces and their Cartesian products are discussed in these two posts (here and here).

A related notion is that of metacompactness. A space is a metacompact space if every open cover has a point-finite open refinement. For a given open cover, any locally finite refinement is a point-finite refinement. Thus paracompactness implies metacompactness. The countable version of metacompactness is also interesting. A space is countably metacompact if every countable open cover has a point-finite open refinement. In fact, for any normal space, the space is countably paracompact if and only of it is countably metacompact (see Corollary 2 below).

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Normal Countably Paracompact Spaces

A good place to begin is to look at countably paracompactness along with normality. In 1951, C. H. Dowker characterized countably paracompactness in the class of normal spaces.

Theorem 1 (Dowker’s Theorem)
Let $X$ be a normal space. The following conditions are equivalent.

The space $X$ is countably paracompact.
Every countable open cover of $X$ has a point-finite open refinement.
If $\left\{U_n: n=1,2,3,\cdots \right\}$ is an open cover of $X$ , there exists an open refinement $\left\{V_n: n=1,2,3,\cdots \right\}$ such that $\overline{V_n} \subset U_n$ for each $n$ .
The product space $X \times Y$ is normal for any compact metric space $Y$ .
The product space $X \times [0,1]$ is normal where $[0,1]$ is the closed unit interval with the usual Euclidean topology.
For each sequence $\left\{A_n \subset X: n=1,2,3,\cdots \right\}$ of closed subsets of $X$ such that $A_1 \supset A_2 \supset A_3 \supset \cdots$ and $\cap_n A_n=\varnothing$ , there exist open sets $B_1,B_2,B_3,\cdots$ such that $A_n \subset B_n$ for each $n$ such that $\cap_n B_n=\varnothing$ .

Dowker’s Theorem is proved in this previous post. Condition 2 in the above formulation of the Dowker’s theorem is not in the Dowker’s theorem in the previous post. In the proof for $1 \rightarrow 2$ in the previous post is essentially $1 \rightarrow 2 \rightarrow 3$ for Theorem 1 above. As a result, we have the following.

Corollary 2
Let $X$ be a normal space. Then $X$ is countably paracompact if and only of $X$ is countably metacompact.

Theorem 1 indicates that normal countably paracompact spaces are important for the discussion of normality in product spaces. As a result of this theorem, we know that normal countably paracompact spaces are productively normal with compact metric spaces. The Cartesian product of normal spaces with compact spaces can be non-normal (an example is found here). When the normal factor is countably paracompact and the compact factor is upgraded to a metric space, the product is always normal. The connection with normality in products is further demonstrated by the following corollary of Theorem 1.

Corollary 3
Let $X$ be a normal space. Let $Y$ be a non-discrete metric space. If $X \times Y$ is normal, then $X$ is countably paracompact.

Since $Y$ is non-discrete, there is a non-trivial convergent sequence (i.e. the sequence represents infinitely many points). Then the sequence along with the limit point is a compact metric subspace of $Y$ . Let’s call this subspace $S$ . Then $X \times S$ is a closed subspace of the normal $X \times Y$ . As a result, $X \times S$ is normal. By Theorem 1, $X$ is countably paracompact.

C. H. Dowker in 1951 raised the question: is every normal space countably paracompact? Put it in another way, is the product of a normal space and the unit interval always a normal space? As a result of Theorem 1, any normal space that is not countably paracompact is called a Dowker space. The search for a Dowker space took about 20 years. In 1955, M. E. Rudin showed that a Dowker space can be constructed from assuming a Souslin line. In the mid 1960s, the existence of a Souslin line was shown to be independent of the usual axioms of set theorey (ZFC). Thus the existence of a Dowker space was known to be consistent with ZFC. In 1971, Rudin constructed a Dowker space in ZFC. Rudin’s Dowker space has large cardinality and is pathological in many ways. Zoltan Balogh constructed a small Dowker space (cardinality continuum) in 1996. Various Dowker space with nicer properties have also been constructed using extra set theory axioms. The first ZFC Dowker space constructed by Rudin is found in [2]. An in-depth discussion of Dowker spaces is found in [3]. Other references on Dowker spaces is found in [4].

Since Dowker spaces are rare and are difficult to come by, we can employ a “probabilistic” argument. For example, any “concrete” normal space (i.e. normality can be shown without using extra set theory axioms) is likely to be countably paracompact. Thus any space that is normal and not paracompact is likely countably paracompact (if the fact of being normal and not paracompact is established in ZFC). Indeed, any well known ZFC example of normal and not paracompact must be countably paracompact. In the long search for Dowker spaces, researchers must have checked all the well known examples! This probability thinking is not meant to be a proof that a given normal space is countably paracompact. It is just a way to suggest a possible answer. In fact, a good exercise is to pick a normal and non-paracompact space and show that it is countably paracompact.

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Some Examples

The following lists out a few classes of spaces that are always countably paracompact.

Metric spaces are countably paracompact.
Paracompact spaces are countably paracompact.
Compact spaces are countably paracompact.
Countably compact spaces are countably paracompact.
Perfectly normal spaces are countably paracompact.
Normal Moore spaces are countably paracompact.
Linearly ordered spaces are countably paracompact.
Shrinking spaces are countably paracompact.

The first four bullet points are clear. Metric spaces are paracompact. It is clear from definition that paracompact spaces, compact and countably compact spaces are countably paracompact. One way to show perfect normal spaces are countably paracompact is to show that they satisfy condition 6 in Theorem 1 (shown here). Any Moore space is perfect (closed sets are $G_\delta$ ). Thus normal Moore space are perfectly normal and hence countably paracompact. The proof of the countably paracompactness of linearly ordered spaces can be found in [1]. See Theorem 5 and Corollary 6 below for the proof of the last bullet point.

As suggested by the probability thinking in the last section, we now look at examples of countably paracompact spaces among spaces that are “normal and not paracompact”. The first uncountable ordinal $\omega_1$ is normal and not paracompact. But it is countably compact and is thus countably paracompact.

Example 1
Any $\Sigma$ -product of uncountably many metric spaces is normal and countably paracompact.

For each $\alpha<\omega_1$ , let $X_\alpha$ be a metric space that has at least two points. Assume that each $X_\alpha$ has a point that is labeled 0. Consider the following subspace of the product space $\prod_{\alpha<\omega_1} X_\alpha$ .

$\displaystyle \Sigma_{\alpha<\omega_1} X_\alpha =\left\{f \in \prod_{\alpha<\omega_1} X_\alpha: \ f(\alpha) \ne 0 \text{ for at most countably many } \alpha \right\}$

The space $\Sigma_{\alpha<\omega_1} X_\alpha$ is said to be the $\Sigma$ -product of the spaces $X_\alpha$ . It is well known that the $\Sigma$ -product of metric spaces is normal, in fact collectionwise normal (this previous post has a proof that $\Sigma$ -product of separable metric spaces is collectionwise normal). On the other hand, any $\Sigma$ -product always contains $\omega_1$ as a closed subset as long as there are uncountably many factors and each factor has at least two points (see the lemma in this previous post). Thus any such $\Sigma$ -product, including the one being discussed, cannot be paracompact.

Next we show that $T=(\Sigma_{\alpha<\omega_1} X_\alpha) \times [0,1]$ is normal. The space $T$ can be reformulated as a $\Sigma$ -product of metric spaces and is thus normal. Note that $T=\Sigma_{\alpha<\omega_1} Y_\alpha$ where $Y_0=[0,1]$ , for any $n$ with $1 \le n<\omega$ , $Y_n=X_{n-1}$ and for any $\alpha$ with $\alpha>\omega$ , $Y_\alpha=X_\alpha$ . Thus $T$ is normal since it is the $\Sigma$ -product of metric spaces. By Theorem 1, the space $\Sigma_{\alpha<\omega_1} X_\alpha$ is countably paracompact. $\square$

Example 2
Let $\tau$ be any uncountable cardinal number. Let $D_\tau$ be the discrete space of cardinality $\tau$ . Let $L_\tau$ be the one-point Lindelofication of $D_\tau$ . This means that $L_\tau=D_\tau \cup \left\{\infty \right\}$ where $\infty$ is a point not in $D_\tau$ . In the topology for $L_\tau$ , points in $D_\tau$ are isolated as before and open neighborhoods at $\infty$ are of the form $L_\tau - C$ where $C$ is any countable subset of $D_\tau$ . Now consider $C_p(L_\tau)$ , the space of real-valued continuous functions defined on $L_\tau$ endowed with the pointwise convergence topology. The space $C_p(L_\tau)$ is normal and not Lindelof, hence not paracompact (discussed here). The space $C_p(L_\tau)$ is also homeomorphic to a $\Sigma$ -product of $\tau$ many copies of the real lines. By the same discussion in Example 1, $C_p(L_\tau)$ is countably paracompact. For the purpose at hand, Example 2 is similar to Example 1. $\square$

Example 3
Consider R. H. Bing’s example G, which is a classic example of a normal and not collectionwise normal space. It is also countably paracompact. This previous post shows that Bing’s Example G is countably metacompact. By Corollary 2, it is countably paracompact. $\square$

Based on the “probabilistic” reasoning discussed at the end of the last section (based on the idea that Dowker spaces are rare), “normal countably paracompact and not paracompact” should be in plentiful supply. The above three examples are a small demonstration of this phenomenon.

Existence of Dowker spaces shows that normality by itself does not imply countably paracompactness. On the other hand, paracompact implies countably paracompact. Is there some intermediate property that always implies countably paracompactness? We point that even though collectionwise normality is intermediate between paracompactness and normality, it is not sufficiently strong to imply countably paracompactness. In fact, the Dowker space constructed by Rudin in 1971 is collectionwise normal.

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More on Countably Paracompactness

Without assuming normality, the following is a characterization of countably paracompact spaces.

Theorem 4
Let $X$ be a topological space. Then the space $X$ is countably paracompact if and only of the following condition holds.

For any decreasing sequence $\left\{A_n: n=1,2,3,\cdots \right\}$ of closed subsets of $X$ such that $\cap_n A_n=\varnothing$ , there exists a decreasing sequence $\left\{B_n: n=1,2,3,\cdots \right\}$ of open subsets of $X$ such that $A_n \subset B_n$ for each $n$ and $\cap_n \overline{B_n}=\varnothing$ .

Proof of Theorem 4
Suppose that $X$ is countably paracompact. Suppose that $\left\{A_n: n=1,2,3,\cdots \right\}$ is a decreasing sequence of closed subsets of $X$ as in the condition in the theorem. Then $\mathcal{U}=\left\{X-A_n: n=1,2,3,\cdots \right\}$ is an open cover of $X$ . Let $\mathcal{V}$ be a locally finite open refinement of $\mathcal{U}$ . For each $n=1,2,3,\cdots$ , define the following:

$B_n=\cup \left\{V \in \mathcal{V}: V \cap A_n \ne \varnothing \right\}$

It is clear that $A_n \subset B_n$ for each $n$ . The open sets $B_n$ are decreasing, i.e. $B_1 \supset B_2 \supset \cdots$ since the closed sets $A_n$ are decreasing. To show that $\cap_n \overline{B_n}=\varnothing$ , let $x \in X$ . The goal is to find $B_j$ such that $x \notin \overline{B_j}$ . Once $B_j$ is found, we will obtain an open set $V$ such that $x \in V$ and $V$ contains no points of $B_j$ .

Since $\mathcal{V}$ is locally finite, there exists an open set $V$ such that $x \in V$ and $V$ meets only finitely many sets in $\mathcal{V}$ . Suppose that these finitely many open sets in $\mathcal{V}$ are $V_1,V_2,\cdots,V_m$ . Observe that for each $i=1,2,\cdots,m$ , there is some $j(i)$ such that $V_i \cap A_{j(i)}=\varnothing$ (i.e. $V_i \subset X-A_{j(i)}$ ). This follows from the fact that $\mathcal{V}$ is a refinement $\mathcal{U}$ . Let $j$ be the maximum of all $j(i)$ where $i=1,2,\cdots,m$ . Then $V_i \cap A_{j}=\varnothing$ for all $i=1,2,\cdots,m$ . It follows that the open set $V$ contains no points of $B_j$ . Thus $x \notin \overline{B_j}$ .

For the other direction, suppose that the space $X$ satisfies the condition given in the theorem. Let $\mathcal{U}=\left\{U_n: n=1,2,3,\cdots \right\}$ be an open cover of $X$ . For each $n$ , define $A_n$ as follows:

$A_n=X-U_1 \cup U_2 \cup \cdots \cup U_n$

Then the closed sets $A_n$ form a decreasing sequence of closed sets with empty intersection. Let $B_n$ be decreasing open sets such that $\bigcap_{i=1}^\infty \overline{B_i}=\varnothing$ and $A_n \subset B_n$ for each $n$ . Let $C_n=X-B_n$ for each $n$ . Then $C_n \subset \cup_{j=1}^n U_j$ . Define $V_1=U_1$ . For each $n \ge 2$ , define $V_n=U_n-\bigcup_{j=1}^{n-1}C_{j}$ . Clearly each $V_n$ is open and $V_n \subset U_n$ . It is straightforward to verify that $\mathcal{V}=\left\{V_n: n=1,2,3,\cdots \right\}$ is a cover of $X$ .

We claim that $\mathcal{V}$ is locally finite in $X$ . Let $x \in X$ . Choose the least $n$ such that $x \notin \overline{B_n}$ . Choose an open set $O$ such that $x \in O$ and $O \cap \overline{B_n}=\varnothing$ . Then $O \cap B_n=\varnothing$ and $O \subset C_n$ . This means that $O \cap V_k=\varnothing$ for all $k \ge n+1$ . Thus the open cover $\mathcal{V}$ is a locally finite refinement of $\mathcal{U}$ . $\square$

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We present another characterization of countably paracompact spaces that involves the notion of shrinkable open covers. An open cover $\mathcal{U}$ of a space $X$ is said to be shrinkable if there exists an open cover $\mathcal{V}=\left\{V(U): U \in \mathcal{U} \right\}$ of the space $X$ such that for each $U \in \mathcal{U}$ , $\overline{V(U)} \subset U$ . If $\mathcal{U}$ is shrinkable by $\mathcal{V}$ , then we also say that $\mathcal{V}$ is a shrinking of $\mathcal{U}$ . Note that Theorem 1 involves a shrinking. Condition 3 in Theorem 1 (Dowker’s Theorem) can rephrased as: every countable open cover of $X$ has a shrinking. This for any normal countably paracompact space, every countable open cover has a shrinking (or is shrinkable).

A space $X$ is a shrinking space if every open cover of $X$ is shrinkable. Every shrinking space is a normal space. This follows from this lemma: A space $X$ is normal if and only if every point-finite open cover of $X$ is shrinkable (see here for a proof). With this lemma, it follows that every shrinking space is normal. The converse is not true. To see this we first show that any shrinking space is countably paracompact. Since any Dowker space is a normal space that is not countably paracompact, any Dowker space is an example of a normal space that is not a shrinking space. To show that any shrinking space is countably paracompact, we first prove the following characterization of countably paracompactness.

Theorem 5
Let $X$ be a space. Then $X$ is countably paracompact if and only of every countable increasing open cover of $X$ is shrinkable.

Proof of Theorem 5
Suppose that $X$ is countably paracompact. Let $\mathcal{U}=\left\{U_1,U_2,U_3,\cdots \right\}$ be an increasing open cover of $X$ . Then there exists a locally open refinement $\mathcal{V}_0$ of $\mathcal{U}$ . For each $n$ , define $V_n=\cup \left\{O \in \mathcal{V}_0: O \subset U_n \right\}$ . Then $\mathcal{V}=\left\{V_1,V_2,V_3,\cdots \right\}$ is also a locally finite refinement of $\mathcal{U}$ . For each $n$ , define

$G_n=\cup \left\{O \subset X: O \text{ is open and } \forall \ m > n, O \cap V_m=\varnothing \right\}$

Let $\mathcal{G}=\left\{G_n: n=1,2,3,\cdots \right\}$ . It follows that $G_n \subset G_m$ if $n<m$ . Then $\mathcal{G}$ is an increasing open cover of $X$ . Observe that for each $n$ , $\overline{G_n} \cap V_m=\varnothing$ for all $m > n$ . Then we have the following:

$\displaystyle \begin{aligned} \overline{G_n}&\subset X-\cup \left\{V_m: m > n \right\} \\&\subset \cup \left\{V_k: k=1,2,\cdots,n \right\} \\&\subset \cup \left\{U_k: k=1,2,\cdots,n \right\}=U_n \end{aligned}$

We have just established that $\mathcal{G}$ is a shrinking of $\mathcal{U}$ , or that $\mathcal{U}$ is shrinkable.

For the other direction, to show that $X$ is countably paracompact, we show that the condition in Theorem 4 is satisfied. Let $\left\{A_1,A_2,A_3,\cdots \right\}$ be a decreasing sequence of closed subsets of $X$ with empty intersection. Then $\mathcal{U}=\left\{U_1,U_2,U_3,\cdots \right\}$ be an open cover of $X$ where $U_n=X-A_n$ for each $n$ . By assumption, $\mathcal{U}$ is shrinkable. Let $\mathcal{V}=\left\{V_1,V_2,V_3,\cdots \right\}$ be a shrinking. We can assume that $\mathcal{V}$ is an increasing sequence of open sets.

For each $n$ , let $B_n=X-\overline{V_n}$ . We claim that $\left\{B_1,B_2,B_3,\cdots \right\}$ is a decreasing sequence of open sets that expand the closed sets $A_n$ and that $\bigcap_{n=1}^\infty \overline{B_n}=\varnothing$ . The expansion part follows from the following:

$A_n=X-U_n \subset X-\overline{V_n}=B_n$

The part about decreasing follows from:

$B_{n+1}=X-\overline{V_{n+1}} \subset X-\overline{V_n}=B_n$

We show that $\bigcap_{n=1}^\infty \overline{B_n}=\varnothing$ . To this end, let $x \in X$ . Then $x \in V_n$ for some $n$ . We claim that $x \notin \overline{B_n}$ . Suppose $x \in \overline{B_n}$ . Since $V_n$ is an open set containing $x$ , $V_n$ must contain a point of $B_n$ , say $y$ . Since $y \in B_n$ , $y \notin \overline{V_n}$ . This in turns means that $y \notin V_n$ , a contradiction. Thus we have $x \notin \overline{B_n}$ as claimed. We have established that every point of $X$ is not in $\overline{B_n}$ for some $n$ . Thus the intersection of all the $\overline{B_n}$ must be empty. We have established the condition in Theorem 4 is satisfied. Thus $X$ is countably paracompact. $\square$

Corollary 6
If $X$ is a shrinking space, then $X$ is countably paracompact.

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Reference

Ball, B. J., Countable Paracompactness in Linearly Ordered Spaces, Proc. Amer. Math. Soc., 5, 190-192, 1954. (link)
Rudin, M. E., A Normal Space $X$ for which $X \times I$ is not Normal, Fund. Math., 73, 179-486, 1971. (link)
Rudin, M. E., Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, (1984) 761-780.
Wikipedia Entry on Dowker Spaces (link)

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$\copyright \ 2016 \text{ by Dan Ma}$

A stroll in Bing’s Example G

Posted on December 5, 2016 by Dan Ma

In this post we take a leisurely walk in Bing’s Example G, which is a classic example of a normal but not collectionwise normal space. Hopefully anyone who is new to this topological space can come away with an intuitive feel and further learn about it. Indeed this is a famous space that had been extensively studied. This example has been written about in several posts in this topology blog. In this post, we explain how Example G is defined, focusing on intuitive idea as much as possible. Of course, the intuitive idea is solely the perspective of the author. Any reader who is interested in building his/her own intuition on this example can skip this post and go straight to the previous introduction. Other blog posts on various subspaces of Example G are here, here and here. Bing’s Example H is discussed here.

At the end of the post, we will demonstrate that the product of Bing’s Example G with the closed unit interval, $F \times [0,1]$ , is a normal space.

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The Product Space Angle

The topology in Example G is tweaked from the product space topology. It is thus a good idea to first examine the relevant product space. Let $P$ be any uncountable set. Let $Q$ be the set of all subsets of $P$ . In other words, $Q$ is the power set of $P$ . Consider the product of $\lvert Q \lvert$ many copies of the two element set $\left\{0,1 \right\}$ . The usual notation of this product space is $2^Q$ . The elements of $2^Q$ are simply the functions from $Q$ into $\left\{0,1 \right\}$ . An arbitrary element of $2^Q$ is a function $f$ that maps every subset of $P$ to either 0 or 1.

Though the base set $P$ can be any uncountable set, it is a good idea to visualize clearly what $P$ is. In the remainder of this section, think of $P$ as the real line $\mathbb{R}$ . Then $Q$ is simply the collection of all subsets of the real line. The elements of the product space are simply functions that map each set of real numbers to either 0 or 1. Or think of each function as a 2-color labeling of the subsets of the real line, where each subset is either red or green for example. There are $2^c$ many subsets of the real line where $c$ is the cardinality of the continuum.

To further visualize the product space, let’s look at a particular subspace of $2^Q$ . For each real number $p$ , define the function $f_p$ such that $f_p$ always maps any set of real numbers that contains $p$ to 1 and maps any set of real numbers that does not contain $p$ to 0. For example, the following are several values of the function $f_0$ .

$f_0([0,1])=1$

$f_0([1,2])=0$

$f_0(\left\{0 \right\})=1$

$f_0(\mathbb{R}-\left\{0 \right\})=0$

$f_0(\mathbb{R})=1$

$f_0(\varnothing)=0$

$f_0(\mathbb{P})=0$

where $\mathbb{P}$ is the set of all irrational numbers. Consider the subspace $F_P=\left\{f_p: p \in P \right\}$ . Members of $F_P$ are easy to describe. Each function in $F_P$ maps a subset of the real line to 0 or 1 depending on whether the subscript belongs to the given subset. Another reason that $F_P$ is important is that Bing’s Example is defined by declaring all points not in $F_P$ isolated points and by allowing all points in $F_P$ retaining the open sets in the product topology.

Any point $f$ in $F_P$ determines $f(q)=0 \text{ or } 1$ based on membership (whether the reference point belongs to the set $q$ ). Points not in $F_P$ have no easy characterization. It seems that any set can be mapped to 0 or 1. Note that any $f$ in $F_P$ maps equally to 0 or 1. So the constant functions $f(q)=0$ and $f(q)=1$ are not in $F_P$ . Furthermore, any $f$ such that $f(q)=1$ for at most countably many $q$ would not be in $F_P$ .

Let’s continue focusing on the product space for the time being. When $F_P$ is considered as a subspace of the product space $2^Q$ , $F_P$ is a discrete space. For each $p \in P$ , there is an open set $W_p$ containing $f_p$ such that $W_p$ contains no other points of $F_P$ . So $F_P$ is relatively discrete in the product space $2^Q$ . Of course $F_P$ cannot be closed in $2^Q$ since $2^Q$ is a compact space. The open set $W_p$ is defined as follows:

$W_p=\left\{f \in 2^Q: f(\left\{p \right\})=1 \text{ and } f(P-\left\{p \right\})=0 \right\}$

It is clear that $f_p \in W_p$ and that $f_t \notin W_p$ for any real number $t \ne p$ .

Two properties of the product space $2^Q$ would be very relevant for the discussion. By the well known Tychonoff theorem, the product space $2^Q$ is compact. Since $P$ is uncountable, $2^Q$ always has the countable chain condition (CCC) since it is the product of separable spaces. A space having CCC means that there can only be at most countably many pairwise disjoint open sets. As a result, the uncountably many open sets $W_p$ cannot be all pairwise disjoint. So there exist at least a pair of $W_p$ , say $W_{a}$ and $W_{b}$ , with nonempty intersection.

The last observation can be generalized. For each $p \in P$ , let $V_p$ be any open set containing $f_p$ (open in the product topology). We observe that there are at least two $a$ and $b$ from $P$ such that $V_a \cap V_b \ne \varnothing$ . If there are only countably many distinct sets $V_p$ , then there are uncountably many $V_p$ that are identical and the observation is valid. So assume that there are uncountably many distinct $V_p$ . By the CCC in the product space, there are at least two $a$ and $b$ with $V_a \cap V_b \ne \varnothing$ . This observation shows that the discrete points in $F_P$ cannot be separated by disjoint open sets. This means that Bing’s Example G is not collectionwise Hausdorff and hence not collectionwise normal.

Another observation is that any disjoint $A_1, A_2 \subset F_P$ can be separated by disjoint open sets. To see this, define the following two open sets $E_1$ and $E_2$ in the product topology.

$q_1=\left\{p \in P: f_p \in A_1 \right\}$

$q_2=\left\{p \in P: f_p \in A_2 \right\}$

$E_1=\left\{f \in 2^Q: f(q_1)=1 \text{ and } f(q_2)=0 \right\}$

$E_2=\left\{f \in 2^Q: f(q_1)=0 \text{ and } f(q_2)=1 \right\}$

It is clear that $A_1 \subset E_1$ and $A_2 \subset E_2$ . Furthermore, $E_1 \cap E_2=\varnothing$ . This observation will be the basis for showing that Bing’s Example G is normal.

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The Topology of Bing’s Example G

The topology for Bing’s Example G is obtained by tweaking the product topology on $2^Q$ . Let $P$ be any uncountable set. Let $Q$ be the set of all subsets of $P$ . The set $F_P$ is defined as above. Bing’s Example G is $F=2^Q$ with points in $F_P$ retaining the open sets in the product topology and with points not in $F_P$ declared isolated. For some reason, in Bing’s original paper, the notation $F$ is used even though the example is identified by G. We will follow Bing’s notation.

The subspace $F_P$ is discrete but not closed in the product topology. However, $F_P$ is both discrete and closed in Bing’s Example G. Based on the discussion in the previous section, one immediate conclusion we can made is that the space $F$ is not collectionwise Hausdorff. This follows from the fact that points in the uncountable closed and discrete set $F_P$ cannot be separated by disjoint open sets. By declaring points not in $F_P$ isolated, the countable chain condition in the original product space $2^Q$ is destroyed. However, there is still a strong trace of CCC around the points in the set $F_P$ , which is sufficient to prevent collectionwise Hausdorffness, and consequently collectionwise normality.

To show that $F$ is normal, let $H$ and $K$ be disjoint closed subsets of $F$ . To make it easy to follow, let $H=A_1 \cup B_1$ and $K=A_2 \cup B_2$ where

$A_1=H \cap F_P \ \ \ \ B_1=H \cap (F-F_P)$

$A_2=K \cap F_P \ \ \ \ B_2=K \cap (F-F_P)$

In other words, $A$ is the non-isolated part and $B$ is the isolated part of the respective closed set. Based on the observation made in the previous section, obtain the disjoint open sets $E_1$ and $E_2$ where $A_1 \subset E_1$ and $A_2 \subset E_2$ . Set the following open sets.

$O_1=(E_1 \cup B_1) - K$

$O_2=(E_2 \cup B_2) - H$

It follows that $O_1$ and $O_2$ are disjoint open sets and that $A_1 \subset O_1$ and $A_2 \subset O_2$ . Thus Bing’s Example G is a normal space.

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Bing’s Example G is Countably Paracompact

We discuss one more property of Bing’s Example G. A space $X$ is countably paracompact if every countable countable open cover of $X$ has a locally finite open refinement. In other words, such a space satisfies the property of being a paracompact space but just for countable open covers. A space is countably metacompact if every countable open cover has a point-finite open refinement (i.e. replacing locally finite in the paracompact definition with point-finite). It is well known that in the class of normal spaces, the two notions are equivalent (see Corollary 2 here). Since Bing’s Example G is normal, we only need to show that it is countably metacompact. Note that Bing’s Example G is not metacompact (see here).

Let $\mathcal{U}$ be a countable open cover of $F$ . Let $\mathcal{U}^*=\left\{U_1,U_2,U_3,\cdots \right\}$ be the set of all open sets in $\mathcal{U}$ that contain points in $F_P$ . For each $i$ , let $A_i=U_i \cap F_P$ . From the perspective of Bing’s Example G, the sets $A_i$ are discrete closed sets. In any normal space, countably many discrete closed sets can be separated by disjoint open sets (see Lemma 1 here). Let $O_1,O_2,O_3,\cdots$ be disjoint open sets such that $A_i \subset O_i$ for each $i$ .

We now build a point-finite open refinement of $\mathcal{U}$ . For each $i$ , let $V_i=U_i \cap O_i$ . Let $V=\cup_{i=1}^\infty V_i$ . Consider the following.

$\mathcal{V}=\left\{V_i: i=1,2,3,\cdots \right\} \cup \left\{\left\{ x \right\}: x \in F-V \right\}$

It follows that $\mathcal{V}$ is an open cover of $F$ . All points of $F_P$ belong to the open sets $V_i$ . Any point that is not in one of the $V_i$ belongs to a singleton open set. It is also clear that $\mathcal{V}$ is a refinement of $\mathcal{U}$ . For each $i$ , $V_i \subset U_i$ and each singleton set is contained in some member of $\mathcal{U}$ . It follows that each point in $F$ belongs to at most finitely many sets in $\mathcal{V}$ . In fact, each point belongs to exactly one set in $\mathcal{V}$ . Each point in $F_P$ belongs to exactly one $V_i$ since the open sets $O_i$ are disjoint. Any point in $V$ belongs to exactly one singleton open set. What we just show is slightly stronger than countably metacompact. The technical term would be countably 1-bounded metacompact.

Since among normal spaces, countably paracompactness is equivalent to countably metacompact, we can now say that Bing’s Example G is a topological space that is normal and countably paracompact. By Dowker’s Theorem, we can conclude that the product of Bing’s Example G with the closed unit interval, $F \times [0,1]$ , is a normal space.

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Previous Posts

Introduction of Bing’s Example G

Some subspaces of Bing’s Example G

Compact subspaces of Bing’s Example G

A subspace of Bing’s Example G

Bing’s Example H

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$\copyright \ 2016 \text{ by Dan Ma}$

Dan Ma's Topology Blog

Expository Articles In General Topology

Tag Archives: Product space

Revisiting example 106 from Steen and Seebach

Lindelof Exercise 2

Lindelof Exercise 1

The product of locally compact paracompact spaces

The product of a perfectly normal space and a metric space is perfectly normal

The product of a normal countably compact space and a metric space is normal

kappa-Dowker space and the first conjecture of Morita

Spaces with shrinking properties

Countably paracompact spaces

A stroll in Bing’s Example G