Perfect preimages of Lindelof spaces

Posted on April 7, 2023 by Dan Ma

Let $f:X \rightarrow Y$ be a mapping from a topological space $X$ onto a topological space $Y$ . If $f$ is a perfect map and $Y$ is a Lindelof space, then so is $X$ . If $f$ is a closed map and $Y$ is a paracompact space, then so is $X$ . In other words, the pre-image of a Lindelof space under a perfect map is always a Lindelof space. Likewise, the pre-image of a paracompact space under a closed map is always a paracompact space. After proving these two facts, we show that for any compact space $Y$ , the product $X \times Y$ is Lindelof (paracompact) for any Lindelof (paracompact) space $X$ . All spaces under consideration are Hausdorff.

Another way to state the above two facts is that Lindelofness is an inverse invariance under perfect maps and that paracompactness is an inverse invariance of the closed maps. In general, a topological property is an inverse invariance of a class of mappings $\mathcal{M}$ if the following holds: for any mapping $f:X \rightarrow Y$ belonging to $\mathcal{M}$ , if $Y$ has the property, then so does $X$ . In contrast, a topological property is an invariance of a class of mappings $\mathcal{M}$ if for any mapping $f:X \rightarrow Y$ belonging to $\mathcal{M}$ , if $X$ has the property, then so does $Y$ .

All mappings under consideration are continuous maps. A mapping $f:X \rightarrow Y$ , where $f(X)=Y$ , is a closed map if for any closed subset $A$ of $X$ , $f(A)$ is closed in $Y$ . A mapping $f:X \rightarrow Y$ , where $f(X)=Y$ , is a perfect map if $f$ is a closed map and that the point inverse $f^{-1}(y)$ is compact for each $y \in Y$ .

Perfect mappings and closed mappings are objects with strong properties. Such a map places a restriction on what topological properties the “domain” space or the “range” space can have. The theorems below indicate that it is not possible to map a non-Lindelof space onto a Lindelof space using a perfect map and that it is not possible to map a non-paracompact space onto a paracompact space using a closed map. On the other hand, it is not possible to map a separable metric space onto a separable but non-metric space using a perfect map (see here). We prove the following theorems.

Theorem 1…. Lindelofness (or the Lindelof property) is an inverse invariant of the perfect maps.

Theorem 2 ….Paracompactness is an inverse invariant of the closed maps.

Lemma 3 …. Let $f:X \longrightarrow Y$ be a closed map with $f(X)=Y$ . Let $V$ be an open subset of $X$ . Define $f_*(V)=\{ y \in Y: f^{-1}(y) \subset V \}$ . Then the set $f_*(V)$ is open in $Y$ and that $f_*(V) \subset f(V)$ .

For the proof of Lemma 3, see Lemma 2 here.

Proof of Theorem 1
Let $f:X \longrightarrow Y$ be a perfect map with $f(X)=Y$ . Suppose $Y$ is Lindelof. Let $\mathcal{U}$ be an open cover of $X$ . Without loss of generality, we can assume that $\mathcal{U}$ is closed under finite unions. For each $U \in \mathcal{U}$ , define $f_*(U)=\{ y \in Y: f^{-1}(y) \subset U \}$ . By Lemma 3, each $f_*(U)$ is an open subset of $Y$ . We claim that $\mathcal{V}=\{ f_*(U): U \in \mathcal{U} \}$ is an open cover of $Y$ . To this end, let $y \in Y$ . Since $f$ is a perfect map, the point inverse $f^{-1}(y)$ is compact. As a result, we can find a finite $\mathcal{F} \subset \mathcal{U}$ such that $f^{-1}(y) \subset \bigcup \mathcal{F}=W$ . Since $\mathcal{U}$ is closed under finite unions, $W \in \mathcal{U}$ . It follows that $y \in f_*(W)$ . Since $Y$ is Lindelof, there exists a countable $\{W_0,W_1,W_2,\cdots \} \subset \mathcal{V}$ such that $Y = \bigcup_{n=0}^\infty W_n$ . For each $n$ , $W_n=f_*(U_n)$ for some $U_n \in \mathcal{U}$ . We claim that $\{U_0,U_1,U_2,\cdots \}$ is a cover of $X$ . To this end, let $x \in X$ . Then for some $n$ , $y=f(x) \in W_n=f_*(U_n)$ . This implies that $x \in f^{-1}(y) \subset U_n$ . Thus, the open cover $\mathcal{U}$ has a countable subcover. This concludes the proof of Theorem 1. $\square$

Proof of Theorem 2
Let $f:X \longrightarrow Y$ be a perfect map with $f(X)=Y$ . Suppose $Y$ is paracompact. Let $\mathcal{U}$ be an open cover of $X$ . For each $U \in \mathcal{U}$ , define $f_*(U)$ as in Lemma 3. By Lemma 3, each $f_*(U)$ is an open subset of $Y$ . Let $\mathcal{V}=\{ f_*(U): U \in \mathcal{U} \}$ . As shown in the proof of Theorem 1, $\mathcal{V}$ is an open cover of $Y$ . Since $Y$ is paracompact, there exists a locally finite open refinement $\mathcal{W}$ of $\mathcal{V}$ . Let $\mathcal{W}_0=\{ f^{-1}(W): W \in \mathcal{W} \}$ .

We show three facts about $\mathcal{W}_0$ . (1) It is an open cover of $X$ . (2) It is a locally finite collection in $X$ . (3) It is a refinement of $\mathcal{U}$ . To see (1), note that $\mathcal{W}$ is an open cover of $Y$ . As a result, $\mathcal{W}_0$ is an open cover of $X$ . To see (2), let $x \in X$ . We find an open $O \subset X$ such that $x \in O$ and such that $O$ intersects only finitely many elements of $\mathcal{W}_0$ . Since $\mathcal{W}$ is locally finite in $Y$ , there exists an open $B \subset Y$ such that $y=f(x) \in B$ and such that $B$ intersects only finitely many elements of $\mathcal{W}$ , say, $W_0,W_1,\cdots,W_n$ . Let $O=f^{-1}(B)$ . Clearly, $x \in O$ . It can be verified that the only elements of $\mathcal{W}_0$ having non-empty intersections with $O$ are $f^{-1}(W_0)$ , $f^{-1}(W_1),\cdots$ , $f^{-1}(W_n)$ . To see (3), let $f^{-1}(W) \in \mathcal{W}_0$ where $W \in \mathcal{W}$ . Then $W \subset V=f_*(U)$ for some $V \in \mathcal{V}$ and some $U \in \mathcal{U}$ . We claim that $f^{-1}(W) \subset U$ . Let $x \in f^{-1}(W)$ . Then $y=f(x) \in W \subset V=f_*(U)$ . This implies that $x \in f^{-1}(y) \subset U$ . It follows that $\mathcal{W}_0$ is a locally finite open refinement of the open cover $\mathcal{U}$ . This completes the proof of Theorem 2. $\square$

Productively Paracompact Spaces

A space $X$ is productively paramcompact if $X \times Y$ is paracompact for every paracompact space $Y$ . The definition for productively Lindelof can be stated in a similar way. For some reason, the term “productively paracompact” is not used in the literature but is a topic that had been extensively studied. It is also a topic found in this site. The following four classes of spaces are productively paracompact (see here and here).

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

The proof for compact spaces being productively paracompact given here uses the Tube Lemma (see here). As applications of Theorem 1 and Theorem 2, we use the two theorems to show that compact spaces are both productively Lindelof and productive paracompact.

Theorem 4…. Let $Y$ any compact space. Then $X \times Y$ is Lindelof for every Lindelof space $X$ .

Theorem 5 ….Let $Y$ any compact space. Then $X \times Y$ is paracompact for every paracompact space $X$ .

Theorems 4 and 5 are corollaries to the Kuratowski theorem (see here) and Theorems 1 and 2 above. Suppose $Y$ is compact. Then the projection map from $X \times Y$ onto $X$ is a closed map. The paracompactness of $X \times Y$ follows whenever $X$ is paracompact. The projection map is also perfect since the point inverses are compact due to the compactness of the factor $Y$ . Then the Lindelofness of $X \times Y$ follows whenever $X$ is Lindelof.

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Making sense of the spaces with small diagonal

Posted on January 26, 2021 by Dan Ma

This is a small attempt toward making sense of the spaces with small diagonal. What makes the property of small diagonal interesting is the longstanding open problem that is discussed below. We use examples to help make sense of the definition of small diagonal. Then we discuss briefly the open problem regarding small diagonal. We discuss the following three examples of compact spaces:

$\omega_1 + 1$ with the ordered topology
The one-point compactification of a discrete space of cardinality $\omega_1$
The double arrow space

By a space we mean a topological space that is Tychonoff, i.e. Hausdorff and completely regular (defined here). Let $X$ be a space. The subset $\Delta=\{ (x,x): x \in X \}$ of the square $X^2=X \times X$ is called the diagonal of the space $X$ .

We want to focus on two diagonal properties. The space $X$ is said to have a $G_\delta$ -diagonal if $\Delta$ is a $G_\delta$ -set in $X^2$ , i.e. $\Delta$ is the intersection of countably many open subsets of $X^2$ . The space $X$ is said to have a small diagonal if for each uncountable subset $A$ of $X^2 \backslash \Delta$ , there is an open subset $O$ of $X^2$ such that $\Delta \subset O$ and $O$ misses uncountably many points of $A$ .

How do these two diagonal properties relate? Any space that has a $G_\delta$ -diagonal also has a small diagonal. This fact can be worked out quite easily based on the definitions. The opposite direction is a totally different matter. In fact, the question of whether having a small diagonal implies having a $G_\delta$ -diagonal is related to a longstanding open question.

Let’s focus on compact spaces. A classic metrization theorem for compact spaces states that any compact space with a $G_\delta$ -diagonal is metrizable (see here). A natural question is: if a compact space has a small diagonal, must it be metrizable? Indeed this is a well known open problem, a longstanding problem that has not completely resolved completely. Before discussion the open problem, we look at the three examples indicated above.

The three examples are all non-metrizable compact spaces and thus do not have a $G_\delta$ -diagonal. We show that they also not have a small diagonal. To appreciate the definition of small diagonal, it is helpful to look spaces that do not have a small diagonal.

Example 1

Consider $X=\omega_1+1$ with the order topology. Using interval notion, $X=[0,\omega_1]$ . This is the space of all countable ordinals $\alpha<\omega_1$ plus the point $\omega_1$ at the end. This is a compact space that is not metrizable. For example, any compact metrizable space would be separable. The last point $\omega_1$ cannot be in the closure of any countable subset. According to the classic theorem mentioned above, $X=[0,\omega_1]$ cannot have a $G_\delta$ -diagonal. We show that it does have a small diagonal too.

Let $A=\{ (\gamma, \gamma+1): \gamma< \omega_1 \}$ . Note that any open set of the form $(\alpha, \omega_1] \times (\alpha, \omega_1]$ contains the point $(\omega_1, \omega_1)$ and contains all but countably many points of $A$ . As a result, any open set containing the diagonal $\Delta$ contains all but countably many points of $A$ . This violates the definition of having a small diagonal. Thus the space $X=[0,\omega_1]$ does not have a small diagonal.

More on the Definition

Example 1 suggests a different angle in looking at the definition. The set $A=\{ (\gamma, \gamma+1): \gamma< \omega_1 \}$ is a $\omega_1$ -length sequence convergent to the diagonal, meaning that any open set containing the diagonal contains a tail of the sequence (in this case containing all but countably many elements in the sequence). The convergent sequence view point is the way Husek defined small diagonal [7].

In [7], $X$ is said to have an $\omega_1$ -accessible diagonal if there is an $\omega_1$ -length sequence $\{ (x_\alpha, y_\alpha) \in X^2 \backslash \Delta: \alpha < \omega_1 \}$ that converges to the diagonal $\Delta$ , meaning that any open set containing $\Delta$ contains all but countably many terms in the sequence. This is exactly what is occurring in the space $X=[0,\omega_1]$ . The set $A=\{ (\gamma, \gamma+1): \gamma< \omega_1 \}$ is precisely a convergent sequence converging to the diagonal. The space $X=[0,\omega_1]$ has an $\omega_1$ -accessible diagonal.

Seen in this light, spaces with a small diagonal are the the spaces that do not have an $\omega_1$ -accessible diagonal (or spaces that have an $\omega_1$ -inaccessible diagonal). Though the definition of Husek is more descriptive, the term small diagonal, suggested by E. van Douwen, has become more popular. The small diagonal defined above is more positive sounding. For example, we say the space $X$ has a small diagonal. In Husek [7], the spaces of interest would be the spaces without an $\omega_1$ -accessible diagonal. This definition is a negative one (defined by the lack of certain thing) and takes more syllables to express. Personally speaking, we prefer the term small diagonal though we understand that $\omega_1$ -accessible diagonal is more descriptive.

A slightly different (but equivalent) way of stating the definition of a space with a small diagonal: a space $X$ has a small diagonal if for any uncountable $A \subset X^2 \backslash \Delta$ , there is an uncountable $B \subset A$ such that $\Delta \cap \overline{B}=\varnothing$ . In the literature, the definition of a space with a small diagonal is usually either this one or the one given at the beginning of this post.

Example 2

Now consider the space $X=\omega_1 \cup \{ \infty \}$ where points in $\omega_1$ are isolated and open neighborhoods of the point $\infty$ are of the form $\{ \infty \} \cup (\omega_1 \backslash F)$ with $F$ being any finite subset of $\omega_1$ . This is usually called the one-point compactification of a discrete space (in this case of size $\omega_1$ ). This space is compact. It is also non-metrizable since it has an uncountable discrete subset. Thus it cannot have a $G_\delta$ -diagonal. The weight of $X$ here is $\omega_1$ . Any compact space whose weight is $\omega_1$ cannot have a small diagonal. This is Fact 1 below.

Some Basic Results

Fact 1
Let $X$ any compact space with $w(X)=\omega_1$ . Then $X$ does not have a small diagonal.

Proof of Fact 1
Let $\mathcal{B}=\{ B_\alpha: \alpha < \omega_1 \}$ be a base for $X$ . Let $\mathcal{B}_1=\{ U \times V: U, V \in \mathcal{B} \}$ , which is a base for $X^2$ . Define $\mathcal{U}=\{ \cup F: F \subset \mathcal{B}_1 \text{ and } \lvert F \lvert < \omega \text{ and } \Delta \subset \cup F \}$ . Since $X$ is compact, the diagonal $\Delta$ is compact. As a result, $\mathcal{U} \ne \varnothing$ . Furthermore, $\lvert \mathcal{U} \lvert=\omega_1$ .

We claim that $\mathcal{U}$ is a base for the diagonal $\Delta$ . To show this, let $W$ be an open subset of $X^2$ such that $\Delta \subset W$ . We can assume that $W$ is the union of elements of the base $\mathcal{B}_1$ . Let $W=\cup \mathcal{A}$ for some $\mathcal{A} \subset \mathcal{B}_1$ . Since $\Delta$ is compact, $\Delta \subset \cup \mathcal{A}_0$ for some finite $\mathcal{A}_0 \subset \mathcal{A}$ . Note that $\cup \mathcal{A}_0 \in \mathcal{U}$ and that $\Delta \subset \cup \mathcal{A}_0 \subset W$ .

Enumerate $\mathcal{U}$ as $\mathcal{U}=\{U_\alpha: \alpha<\omega_1 \}$ . Since $X$ is compact and has uncountable weight, $X$ is not metrizable. Hence the diagonal $\Delta$ is not a $G_\delta$ -set. As a result, for each $\alpha<\omega_1$ , $(\bigcap_{\beta<\alpha} U_\beta) \backslash \Delta \ne \varnothing$ . Furthermore, for any countable $\{ y_0, y_1, y_2, \cdots \} \subset X^2 \backslash \Delta$ , $[(\bigcap_{\beta<\alpha} U_\beta) \cap (\bigcap_{n \in \omega} X^2 \backslash \{ y_n \}) ]\backslash \Delta \ne \varnothing$ .

Pick $x_0 \in U_0$ . For any $\alpha<\omega_1$ with $\alpha > 0$ , choose $x_\alpha \in (\bigcap_{\beta<\alpha} U_\beta) \backslash (\Delta \cup \{x_\delta: \delta < \alpha \})$ .

Then the sequence $\{ x_\alpha: \alpha < \omega_1 \}$ converges to the diagonal $\Delta$ . To see this, fix $U_\gamma \in \mathcal{U}$ . From the way the sequence is chosen, $x_\alpha \in U_\gamma$ for all $\alpha > \gamma$ . This concludes the proof that $X$ does not have a small diagonal. $\square$

Fact 2
Let $X$ any compact space with $w(X) \le \omega_1$ . Then if $X$ has a small diagonal, then $w(X)=\omega$ and thus $X$ is metrizable.

Fact 2 is an easy corollary of Fact 1. Fact 2 says that any compact space with “small” weight (no more than $\omega_1$ ) is metrizable if it has a small diagonal. Both Example 1 and Example 2 are compact non-metrizable spaces with small weight. Therefore they do not have small diagonal. The following basic fact is also useful.

Fact 3
Let $X$ any compact space. Then if $X$ has a small diagonal, then $t(X)=\omega$ , i.e. $X$ is countably tight.

Proof of Fact 3
Suppose $X$ is uncountably tight. We show that it does not have a small diagonal. A sequence of points $\{ x_\alpha \in X: \alpha< \tau \}$ is a free sequence of length $\tau$ if for each $\alpha< \tau$ , $\overline{\{ x_\gamma: \gamma < \alpha \}} \cap \overline{\{ x_\gamma: \gamma \ge \alpha \}}=\varnothing$ . For any compact space with tightness at least $\omega_1$ , there exists a free sequence of length $\omega_1$ (see Lemma 2 here). Thus in the space $X$ in question, there exists a free sequence $\{ x_\alpha \in X: \alpha< \omega_1 \}$ .

The main result in [9] says that if a compact space contains a free sequence of length $\omega_1$ , then it contains a free sequence of the same length that is convergent. We now assume that the above free sequence $\{ x_\alpha \in X: \alpha< \omega_1 \}$ is also convergent, i.e. it converges to some point $x \in X$ . Thus every open set containing $x$ contains all but countably many $x_\alpha$ . Consider the sequence $\{ (x_\alpha, x_{\alpha+1}) \in X^2: \alpha< \omega_1 \}$ . Observe that every open set in $X^2$ containing the point $(x,x)$ contains all but countably pairs $(x_\alpha, x_{\alpha+1})$ . This implies that every open set containing the diagonal $\Delta$ contains all but countably many points in the sequence. This shows that the compact $X$ does not have a small diagonal. $\square$

According to Fact 3, the compact ordinal $\omega_1+1=[0, \omega_1]$ does not have a small diagonal since it is uncountably tight at the last point $\omega_1$ .

Example 3

We now consider the double arrow space. Let $D=[0,1] \times \{0, 1 \}$ . The space $D$ consists of two copies of the unit interval, an upper one and a lower one. See the first two diagrams in this previous post. For $0 \le a <1$ , a basic open set containing the point $(a, 1)$ in the upper interval is of the form $\biggl( [a,b) \times \{ 1 \} \biggr) \cup \biggl( (a,b) \times \{0 \} \biggr)$ . For $0<a \le 1$ , a basic open set containing the point $(a, 0)$ in the lower interval is of the form $\biggl( (c,a) \times \{ 1 \} \biggr) \cup \biggl( (c,a] \times \{0 \} \biggr)$ . The rightmost point in the upper interval $(1,1)$ and the leftmost point in the lower interval $(0,0)$ are made isolated points.

The double arrow space $D$ is compact, perfectly normal and not metrizable (discussed here). Thus $D$ does not have a $G_\delta$ -diagonal. We do not have a direct way of showing that it does not have a small diagonal. We rely on a result from [5], which says that every compact metrizably fibered space with a small diagonal is metrizable. In light of this result, we only need to show that the double arrow space $D$ is metrizably fibered. A space $Y$ is metrizably fibered if there is a continuous map from $Y$ onto some metrizable space $M$ such that each point inverse is metrizable.

Starting with the double arrow space $D$ , let $f:D \rightarrow [0,1]$ be defined by $f((a,0))=f((a,1))=a$ for each $0 \le a \le 1$ . This is a two-to-one continuous map from the double arrow space $D$ onto the unit interval. The map $f$ is essentially a quotient map, the result of identifying $\{ (a,0), (a, 1) \}$ as one point $a$ . By the result in [5], the double arrow space $D$ cannot have a small diagonal.

The Open Problem

As mentioned earlier, what makes the property of small diagonal is a related longstanding open problem. The statement “every compact space with a $G_\delta$ -diagonal is metrizable” is true in ZFC. What about the following statement?

(*) Every compact space with a small diagonal is metrizable.

This question whether this statement is true was raised in Husek [7]. As of the writing of this article, the problem is still unsolved. There are partial results, some consistent results and some ZFC results.

Assuming CH, the answer to Hesek’s question is positive. Husek [7] showed that under CH, every compact space $X$ with a small diagonal such that the tightness of $X$ is countable is metrizable. Fact 3 from above states that every compact space with a small diagonal has countble tightness. Combining the two results, it follows that under CH any compact space with a small diagonal is metrizable. Fact 3 followed from a result by Juhasz and Szentmikloss [9]. Dow and Pavlov [3] showed that under PFA, every compact space with a small diagonal is metrizable.

There are partial answers in ZFC. We mention three results. The first one is Fact 2 discussed above, which is that compact spaces with small diagonal are metrizable if there is a weight restriction (weight no more than $\omega_1$ ). If there exists a non-metrizable compact space with a small diagonal, its weight would have to be greater than $\omega_1$ .

Gruenhage [5] showed that every compact metrizably fibered space with a small diagonal is metrizable (this result is discussed in Example 3 above). Dow and Hart [2] showed that every compact space with a small diagonal that is weight $\omega_1$ fibered is metrizable. The notion of weight $\omega_1$ fibered is a generalization of metrizably fibered. A space $X$ is weight $\omega_1$ fibered if there is a continuous surjection $f: X \rightarrow Y$ such that $Y$ and each point inverse $f^{-1}(y)$ have weight at most $\omega_1$ .

The statement “every countably compact space with a $G_\delta$ -diagonal is metrizable” is a true statement in ZFC (see here). How about the statement “every countably compact space with a small diagonal is metrizable”, the statement (*) above with compact replaced by countably compact? Gruenhage [5] showed that this statement is consistent with and independent of ZFC. To prove or disprove this statement extra set theory assumptions beyond ZFC are required. This provides an interesting contrast between compact and countably compact with respect to the open problem. By broadening Husek’s question from compact to countably compact, the statement cannot be settled in ZFC. Dow and Pavlov [3] provided two consistent examples of “countably compact non-metrizable with small diagonal” that are improvements upon examples from Gruenhage.

Now consider the statement “Lindelof space with a small diagonal must have a $G_\delta$ -diagonal.” Dow and Pavlov [3] provided a consistent negative answer, an example of a Lindelof space with a small diagonal that does not have a $G_\delta$ -diagonal (under negation of CH).

Another broad natural question is: what do compact spaces with small diagonal look like? The main problem is, of course, trying to see if these spaces are metrizable. There have been attempts to explore this general question of what these spaces look like. According to Juhasz and Szentmikloss [9], these spaces have countable tightness (Fact 2 above). However, it is not known if compact spaces with small diagonal have points of countable character. Dow and Hart [2] uncovered a surprising connection that if there is a subset of the real line that is a Luzin set, every compact space with a small diagonal does have points of countable character. Dow and Hart in the same paper also showed that in every compact space with a small diagonal, CCC subspaces have countable $\pi$ -weight.

This is a brief walk through of the open problem based on the statement (*) indicated above. To find out more, consult with the references listed below. Beyond the main open problem, there are many angles to be explored.

Reference

Arhangelskii, A., Bella, A., Few observations on topological spaces with small diagonal, Zb. Rad. Filoz. Fak. Nisu, 6, No. 2, 211-213, 1992.
Dow, A., Hart, P. Elementary chains and compact spaces with a small diagonal, Indagationes Mathematicae, 23, No. 3, 438-447, 2012.
Dow, A., Pavlov, O. More about spaces with a small diagonal, Fund. Math., 191, No. 1, 67-80, 2006.
Dow, A., Pavlov, O. Perfect preimages and small diagonal, Topology Proc., 31, No. 1, 89-95, 2007.
Gruenhage, G., Spaces having a small diagonal, Topology Appl., 22, 183-200, 2002.
Gruenhage, G., Generalized metrizable spaces, Recent Progress in General Topology III (K.P. Hart, J. van Mill, and P. Simon, eds.), Atlantis Press 2014.
Husek, M., Topological spaces without $\kappa$ -accessible diagonal, Comment. Math. Univ. Carolin., 18, No. 4, 777-788, 1977.
Juhasz, I., Cardinals Functions II, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 63-109, 1984.
Juhasz, I., Szentmikloss, Z. Convergent free sequences in compact spaces, Proc. Amer. Math. Soc., 116, No. 4, 1153-1160, 1992.
Zhou, H. X., On The Small Diagonals, Topology Appl., 13, 283-293, 1982.

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Helly Space

Posted on June 10, 2019 by Dan Ma

This is a discussion on a compact space called Helly space. The discussion here builds on the facts presented in Counterexample in Topology [2]. Helly space is Example 107 in [2]. The space is named after Eduard Helly.

Let $I=[0,1]$ be the closed unit interval with the usual topology. Let $C$ be the set of all functions $f:I \rightarrow I$ . The set $C$ is endowed with the product space topology. The usual product space notation is $I^I$ or $\prod_{t \in I} W_t$ where each $W_t=I$ . As a product of compact spaces, $C=I^I$ is compact.

Any function $f:I \rightarrow I$ is said to be increasing if $f(x) \le f(y)$ for all $x<y$ (such a function is usually referred to as non-decreasing). Helly space is the subspace $X$ consisting of all increasing functions. This space is Example 107 in Counterexample in Topology [2]. The following facts are discussed in [2].

The space $X$ is compact.
The space $X$ is first countable (having a countable base at each point).
The space $X$ is separable.
The space $X$ has an uncountable discrete subspace.

From the last two facts, Helly space is a compact non-metrizable space. Any separable metric space would have countable spread (all discrete subspaces must be countable).

The compactness of $X$ stems from the fact that $X$ is a closed subspace of the compact space $C$ .

Further Discussion

Additional facts concerning Helly space are discussed.

The product space $\omega_1 \times X$ is normal.
Helly space $X$ contains a copy of the Sorgenfrey line.
Helly space $X$ is not hereditarily normal.

The space $\omega_1$ is the space of all countable ordinals with the order topology. Recall $C$ is the product space $I^I$ . The product space $\omega_1 \times C$ is Example 106 in [2]. This product is not normal. The non-normality of $\omega_1 \times C$ is based on this theorem: for any compact space $Y$ , the product $\omega_1 \times Y$ is normal if and only if the compact space $Y$ is countably tight. The compact product space $C$ is not countably tight (discussed here). Thus $\omega_1 \times C$ is not normal. However, the product $\omega_1 \times X$ is normal since Helly space $X$ is first countable.

To see that $X$ contains a copy of the Sorgenfrey line, consider the functions $h_t:I \rightarrow I$ defined as follows:

$\displaystyle h_t(x) = \left\{ \begin{array}{ll} \displaystyle 0 &\ \ \ \ \ \ 0 \le x \le t \\ \text{ } & \text{ } \\ \displaystyle 1 &\ \ \ \ \ \ t<x \le 1 \\ \end{array} \right.$

for all $0<t<1$ . Let $S=\{ h_t: 0<t<1 \}$ . Consider the mapping $\gamma: (0,1) \rightarrow S$ defined by $\gamma(t)=h_t$ . With the domain $(0,1)$ having the Sorgenfrey topology and with the range $S$ being a subspace of Helly space, it can be shown that $\gamma$ is a homeomorphism.

With the Sorgenfrey line $S$ embedded in $X$ , the square $X \times X$ contains a copy of the Sorgenfrey plane $S \times S$ , which is non-normal (discussed here). Thus the square of Helly space is not hereditarily normal. A more interesting fact is that Helly space is not hereditarily normal. This is discussed in the next section.

Finding a Non-Normal Subspace of Helly Space

As before, $C$ is the product space $I^I$ where $I=[0,1]$ and $X$ is Helly space consisting of all increasing functions in $C$ . Consider the following two subspaces of $X$ .

$Y_{0,1}=\{ f \in X: f(I) \subset \{0, 1 \} \}$

$Y=X - Y_{0,1}$

The subspace $Y_{0,1}$ is a closed subset of $X$ , hence compact. We claim that subspace $Y$ is separable and has a closed and discrete subset of cardinality continuum. This means that the subspace $Y$ is not a normal space.

First, we define a discrete subspace. For each $x$ with $0<x<1$ , define $f_x: I \rightarrow I$ as follows:

$\displaystyle f_x(y) = \left\{ \begin{array}{ll} \displaystyle 0 &\ \ \ \ \ \ 0 \le y < x \\ \text{ } & \text{ } \\ \displaystyle \frac{1}{2} &\ \ \ \ \ y=x \\ \text{ } & \text{ } \\ \displaystyle 1 &\ \ \ \ \ \ x<y \le 1 \\ \end{array} \right.$

Let $H=\{ f_x: 0<x<1 \}$ . The set $H$ as a subspace of $X$ is discrete. Of course it is not discrete in $X$ since $X$ is compact. In fact, for any $f \in Y_{0,1}$ , $f \in \overline{H}$ (closure taken in $X$ ). However, it can be shown that $H$ is closed and discrete as a subset of $Y$ .

We now construct a countable dense subset of $Y$ . To this end, let $\mathcal{B}$ be a countable base for the usual topology on the unit interval $I=[0,1]$ . For example, we can let $\mathcal{B}$ be the set of all open intervals with rational endpoints. Furthermore, let $A$ be a countable dense subset of the open interval $(0,1)$ (in the usual topology). For convenience, we enumerate the elements of $A$ and $\mathcal{B}$ .

$A=\{ a_1,a_2,a_3,\cdots \}$

$\mathcal{B}=\{B_1,B_2,B_3,\cdots \}$

We also need the following collections.

$\mathcal{G}=\{G \subset \mathcal{B}: G \text{ is finite and is pairwise disjoint} \}$

$\mathcal{A}=\{F \subset A: F \text{ is finite} \}$

For each $G \in \mathcal{G}$ and for each $F \in \mathcal{A}$ with $\lvert G \lvert=\lvert F \lvert=n$ , we would like to arrange the elements in increasing order, notated as follow:

$F=\{t_1,t_2,\cdots,t_n \}$

$G=\{E_1,E_2,\cdots,E_n \}$

For the set $F$ , we have $0<t_1<t_2< \cdots <t_n<1$ . For the set $G$ , $E_i$ is to the left of $E_j$ for $i<j$ . Note that elements of $G$ are pairwise disjoint. Furthermore, write $E_i=(p_i,q_i)$ . If $0 \in E_1$ , then $E_1=[p_1,q_1)=[0,q_1)$ . If $1 \in E_n$ , then $E_n=(p_n,q_n]=(p_n,1]$ .

For each $F$ and $G$ as detailed above, we define a function $L(F,G):I \rightarrow I$ as follows:

$\displaystyle L(F,G)(x) = \left\{ \begin{array}{ll} \displaystyle t_1 &\ \ \ \ \ 0 \le x < q_1 \\ \text{ } & \text{ } \\ \displaystyle t_2 &\ \ \ \ \ q_1 \le x < q_2 \\ \text{ } & \text{ } \\ \displaystyle \vdots &\ \ \ \ \ \vdots \\ \text{ } & \text{ } \\ \displaystyle t_{n-1} &\ \ \ \ \ q_{n-2} \le x < q_{n-1} \\ \text{ } & \text{ } \\ \displaystyle t_n &\ \ \ \ \ q_{n-1} \le x \le 1 \\ \end{array} \right.$

The following diagram illustrates the definition of $L(F,G)$ when both $F$ and $G$ have 4 elements.

Figure 1 – Member of a countable dense set

Let $D$ be the set of $L(F,G)$ over all $F \in \mathcal{A}$ and $G \in \mathcal{G}$ . The set $D$ is a countable set. It can be shown that $D$ is dense in the subspace $Y$ . In fact $D$ is dense in the entire Helly space $X$ .

To summarize, the subspace $Y$ is separable and has a closed and discrete subset of cardinality continuum. This means that $Y$ is not normal. Hence Helly space $X$ is not hereditarily normal. According to Jones’ lemma, in any normal separable space, the cardinality of any closed and discrete subspace must be less than continuum (discussed here).

Remarks

The preceding discussion shows that both Helly space and the square of Helly space are not hereditarily normal. This is actually not surprising. According to a theorem of Katetov, for any compact non-metrizable space $V$ , the cube $V^3$ is not hereditarily normal (see Theorem 3 in this post). Thus a non-normal subspace is found in $V$ , $V \times V$ or $V \times V \times V$ . In fact, for any compact non-metric space $V$ , an excellent exercise is to find where a non-normal subspace can be found. Is it in $V$ , the square of $V$ or the cube of $V$ ? In the case of Helly space $X$ , a non-normal subspace can be found in $X$ .

A natural question is: is there a compact non-metric space $V$ such that both $V$ and $V \times V$ are hereditarily normal and $V \times V \times V$ is not hereditarily normal? In other words, is there an example where the hereditarily normality fails at dimension 3? If we do not assume extra set-theoretic axioms beyond ZFC, any compact non-metric space $V$ is likely to fail hereditarily normality in either $V$ or $V \times V$ . See here for a discussion of this set-theoretic question.

Reference

Kelly, J. L., General Topology, Springer-Verlag, New York, 1955.
Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.

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Looking for spaces in which every compact subspace is metrizable

Posted on June 11, 2017 by Dan Ma

Once it is known that a topological space is not metrizable, it is natural to ask, from a metrizability standpoint, which subspaces are metrizable, e.g. whether every compact subspace is metrizable. This post discusses several classes of spaces in which every compact subspace is metrizable. Though the goal here is not to find a complete characterization of such spaces, this post discusses several classes of spaces and various examples that have this property. The effort brings together many interesting basic and well known facts. Thus the notion “every compact subspace is metrizable” is an excellent learning opportunity.

Several Classes of Spaces

The notion “every compact subspace is metrizable” is a very broad class of spaces. It includes well known spaces such as Sorgenfrey line, Michael line and the first uncountable ordinal $\omega_1$ (with the order topology) as well as Moore spaces. Certain function spaces are in the class “every compact subspace is metrizable”. The following diagram is a good organizing framework.

$\displaystyle \begin{aligned} &1. \ \text{Metrizable} \\&\ \ \ \ \ \ \ \ \ \Downarrow \\&2. \ \text{Submetrizable} \Longleftarrow 5. \ \exists \ \text{countable network} \\&\ \ \ \ \ \ \ \ \ \Downarrow \\&3. \ \exists \ G_\delta \text{ diagonal} \\&\ \ \ \ \ \ \ \ \ \Downarrow \\&4. \ \text{Every compact subspace is metrizable} \end{aligned}$

Let $(X, \tau)$ be a space. It is submetrizable if there is a topology $\tau_1$ on the set $X$ such that $\tau_1 \subset \tau$ and $(X, \tau_1)$ is a metrizable space. The topology $\tau_1$ is said to be weaker (coarser) than $\tau$ . Thus a space $X$ is submetrizable if it has a weaker metrizable topology.

Let $\mathcal{N}$ be a set of subsets of the space $X$ . $\mathcal{N}$ is said to be a network for $X$ if for every open subset $O$ of $X$ and for each $x \in O$ , there exists $N \in \mathcal{N}$ such that $x \in N \subset O$ . Having a network that is countable in size is a strong property (see here for a discussion on spaces with a countable network).

The diagonal of the space $X$ is the subset $\Delta=\left\{(x,x): x \in X \right\}$ of the square $X \times X$ . The space $X$ has a $G_\delta$ -diagonal if $\Delta$ is a $G_\delta$ -subset of $X \times X$ , i.e. $\Delta$ is the intersection of countably many open subsets of $X \times X$ .

The implication $1 \Longrightarrow 2$ is clear. For $5 \Longrightarrow 2$ , see Lemma 1 in this previous post on countable network. The implication $2 \Longrightarrow 3$ is left as an exercise. To see $3 \Longrightarrow 4$ , let $K$ be a compact subset of $X$ . The property of having a $G_\delta$ -diagonal is hereditary. Thus $K$ has a $G_\delta$ -diagonal. According to a well known result, any compact space with a $G_\delta$ -diagonal is metrizable (see here).

None of the implications in the diagram is reversible. The first uncountable ordinal $\omega_1$ is an example for $4 \not \Longrightarrow 3$ . This follows from the well known result that any countably compact space with a $G_\delta$ -diagonal is metrizable (see here). The Mrowka space is an example for $3 \not \Longrightarrow 2$ (see here). The Sorgenfrey line is an example for both $2 \not \Longrightarrow 5$ and $2 \not \Longrightarrow 1$ .

To see where the examples mentioned earlier are placed, note that Sorgenfrey line and Michael line are submetrizable, both are submetrizable by the usual Euclidean topology on the real line. Each compact subspace of the space $\omega_1$ is countable and is thus contained in some initial segment $[0,\alpha]$ which is metrizable. Any Moore space has a $G_\delta$ -diagonal. Thus compact subspaces of a Moore space are metrizable.

Function Spaces

We now look at some function spaces that are in the class “every compact subspace is metrizable.” For any Tychonoff space (completely regular space) $X$ , $C_p(X)$ is the space of all continuous functions from $X$ into $\mathbb{R}$ with the pointwise convergence topology (see here for basic information on pointwise convergence topology).

Theorem 1
Suppose that $X$ is a separable space. Then every compact subspace of $C_p(X)$ is metrizable.

Proof
The proof here actually shows more than is stated in the theorem. We show that $C_p(X)$ is submetrizable by a separable metric topology. Let $Y$ be a countable dense subspace of $X$ . Then $C_p(Y)$ is metrizable and separable since it is a subspace of the separable metric space $\mathbb{R}^{\omega}$ . Thus $C_p(Y)$ has a countable base. Let $\mathcal{E}$ be a countable base for $C_p(Y)$ .

Let $\pi:C_p(X) \longrightarrow C_p(Y)$ be the restriction map, i.e. for each $f \in C_p(X)$ , $\pi(f)=f \upharpoonright Y$ . Since $\pi$ is a projection map, it is continuous and one-to-one and it maps $C_p(X)$ into $C_p(Y)$ . Thus $\pi$ is a continuous bijection from $C_p(X)$ into $C_p(Y)$ . Let $\mathcal{B}=\left\{\pi^{-1}(E): E \in \mathcal{E} \right\}$ .

We claim that $\mathcal{B}$ is a base for a topology on $C_p(X)$ . Once this is established, the proof of the theorem is completed. Note that $\mathcal{B}$ is countable and elements of $\mathcal{B}$ are open subsets of $C_p(X)$ . Thus the topology generated by $\mathcal{B}$ is coarser than the original topology of $C_p(X)$ .

For $\mathcal{B}$ to be a base, two conditions must be satisfied – $\mathcal{B}$ is a cover of $C_p(X)$ and for $B_1,B_2 \in \mathcal{B}$ , and for $f \in B_1 \cap B_2$ , there exists $B_3 \in \mathcal{B}$ such that $f \in B_3 \subset B_1 \cap B_2$ . Since $\mathcal{E}$ is a base for $C_p(Y)$ and since elements of $\mathcal{B}$ are preimages of elements of $\mathcal{E}$ under the map $\pi$ , it is straightforward to verify these two points. $\square$

Theorem 1 is actually a special case of a duality result in $C_p$ function space theory. More about this point later. First, consider a corollary of Theorem 1.

Corollary 2
Let $X=\prod_{\alpha<c} X_\alpha$ where $c$ is the cardinality continuum and each $X_\alpha$ is a separable space. Then every compact subspace of $C_p(X)$ is metrizable.

The key fact for Corollary 2 is that the product of continuum many separable spaces is separable (this fact is discussed here). Theorem 1 is actually a special case of a deep result.

Theorem 3
Suppose that $X=\prod_{\alpha<\kappa} X_\alpha$ is a product of separable spaces where $\kappa$ is any infinite cardinal. Then every compact subspace of $C_p(X)$ is metrizable.

Theorem 3 is a much more general result. The product of any arbitrary number of separable spaces is not separable if the number of factors is greater than continuum. So the proof for Theorem 1 will not work in the general case. This result is Problem 307 in [2].

A Duality Result

Theorem 1 is stated in a way that gives the right information for the purpose at hand. A more correct statement of Theorem 1 is: $X$ is separable if and only if $C_p(X)$ is submetrizable by a separable metric topology. Of course, the result in the literature is based on density and weak weight.

The cardinal function of density is the least cardinality of a dense subspace. For any space $Y$ , the weight of $Y$ , denoted by $w(Y)$ , is the least cardinaility of a base of $Y$ . The weak weight of a space $X$ is the least $w(Y)$ over all space $Y$ for which there is a continuous bijection from $X$ onto $Y$ . Thus if the weak weight of $X$ is $\omega$ , then there is a continuous bijection from $X$ onto some separable metric space, hence $X$ has a weaker separable metric topology.

There is a duality result between density and weak weight for $X$ and $C_p(X)$ . The duality result:

The density of $X$ coincides with the weak weight of $C_p(X)$ and the weak weight of $X$ coincides with the density of $C_p(X)$ . These are elementary results in $C_p$ -theory. See Theorem I.1.4 and Theorem I.1.5 in [1].

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Reference

Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
Tkachuk V. V., A $C_p$ -Theory Problem Book, Topological and Function Spaces, Springer, New York, 2011.

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$\copyright$ 2017 – 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

An exercise gleaned from the proof of a theorem on pseudocompact space

Posted on August 2, 2015 by Dan Ma

Filling in the gap is something that is done often when following a proof in a research paper or other published work. In fact this is necessary since it is not feasible for authors to prove or justify every statement or assertion in a proof (or define every term). The gap could be a basic result or could be an older result from another source. If the gap is a basic result or a basic fact that is considered folklore, it may be OK to put it on hold in the interest of pursuing the main point. Then come back later to fill the gap. In any case, filling in gaps is a great learning opportunity. In this post, we focus on one such example of filling in the gap. The example is from the book called Topological Function Spaces by A. V. Arkhangelskii [1].

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Pseudocompactness

The exercise we wish to highlight deals with continuous one-to-one functions defined on pseudocompact spaces. We first give a brief backgrounder on pseudocompact spaces with links to earlier posts.

All spaces considered are Hausdorff spaces. A space $X$ is a pseudocompact space if every continuous real-valued function defined on $X$ is bounded, i.e., if $f:X \rightarrow \mathbb{R}$ is a continuous function, then $f(X)$ is a bounded set in the real line. Compact spaces are pseudocompact. In fact, it is clear from definitions that

$\text{compact} \Longrightarrow \text{countably compact} \Longrightarrow \text{pseudocompact}$

None of the implications can be reversed. An example of a pseudocompact space that is not countably compact is the space $\Psi(\mathcal{A})$ where $\mathcal{A}$ is a maximal almost disjoint family of subsets of $\omega$ (see here for the details). Some basic results on pseudocompactness focus on the conditions to add in order to turn a pseudocompact space into countably compact or even compact. For example, for normal spaces, pseudocompact implies countably compact. This tells us that when looking for pseudocompact space that is not countably compact, do not look among normal spaces. Another interesting result is that pseudocompact + metacompact implies compact. Likewise, when looking for pseudocompact space that is not compact, look among non-metacompact spaces. On the other hand, this previous post discusses when a pseudocompact space is metrizable. Another two previous posts also discuss pseudocompactness (see here and here).

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The exercise

Consider Theorem II.6.2 part (c) in pp. 76-77 in [1]. We do not state the theorem because it is not the focus here. Instead, we focus on an assertion in the proof of Theorem II.6.2.

The exercise that we wish to highlight is stated in Theorem 2 below. Theorem 1 is a standard result about continuous one-to-one functions defined on compact spaces and is stated here to contrast with Theorem 2.

Theorem 1
Let $Y$ be a compact space. Let $g: Y \rightarrow Z$ be a one-to-one continuous function from $Y$ onto a space $Z$ . Then $g$ is a homeomorphism.

Theorem 2
Let $Y$ be a pseudocompact space. Let $g: Y \rightarrow Z$ be a one-to-one continuous function from $Y$ onto $Z$ where $Z$ is a separable and metrizable space. Then $g$ is a homeomorphism.

Theorem 1 says that any continuous one-to-one map from a compact space onto another compact space is a homeomorphism. To show a given map between two compact spaces is a homeomorphism, we only need to show that it is continuous in one direction. Theorem 2, the statement used in the proof of Theorem II.6.2 in [1], says that the standard result for compact spaces can be generalized to pseudocompactness if the range space is nice.

The proof of Theorem II.6.2 part (c) in [1] quoted [2] as a source for the assertion in our Theorem 2. Here, we leave both Theorem 1 and Theorem 2 as exercise. One way to prove Theorem 2 is to show that whenever there exists a map $g$ as described in Theorem 2, the domain $Y$ must be compact. Then Theorem 1 will finish the job.

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Reference

Arkhangelskii A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
Arkhangelskii A. V., Ponomarev V. I., Fundamental of general topology: problems and exercises, Reidel, 1984. (Translated from the Russian).

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$\copyright \ 2015 \text{ by Dan Ma}$

Products of compact spaces with countable tightness

Posted on July 1, 2015 by Dan Ma

In the previous two posts, we discuss the definitions of the notion of tightness and its relation with free sequences. This post and the next post are to discuss the behavior of countable tightness under the product operation. In this post, we show that countable tightness behaves well in products of compact space. In particular we show that countable tightness is preserved in finite products and countable products of compact spaces. In the next post we show that countable tightness is easily destroyed in products of sequential fans and that the tightness of such a product can be dependent on extra set theory assumptions. All spaces are Hausdorff and regular.

The following theorems are the main results in this post.

Theorem 1
Let $X$ and $Y$ be countably tight spaces. If one of $X$ and $Y$ is compact, then $X \times Y$ is countably tight.

Theorem 2
The product of finitely many compact countably tight spaces is countably tight.

Theorem 3
Suppose that $X_1, X_2, X_3, \cdots$ are countably many compact spaces such that each $X_i$ has at least two points. If each $X_i$ is a countably tight space, then the product space $\prod_{i=1}^\infty X_i$ is countably tight.

Finite products

Before proving Theorem 1 and Theorem 2, we prove the following results.

A Fact About Continuity
Let $f:X \rightarrow Y$ be a continuous map from a space $X$ onto a space $Y$ . If $B \subset X$ , $x \in \overline{B}$ , and $y=f(x)$ , then $y \in \overline{f(B)}$ .

Theorem 4
Let $f:Y_1 \rightarrow Y_2$ be a continuous and closed map from the space $Y_1$ onto the space $Y_2$ . Suppose that the space $Y_2$ is countably tight and that each fiber of the map $f$ is countably tight. Then the space $Y_1$ is countably tight.

Proof of Theorem 4
Let $x \in Y_1$ and $x \in \overline{A}$ where $A \subset Y_1$ . We proceed to find a countable $W \subset Y_1$ such that $x \in \overline{W}$ . Choose $y \in Y_2$ such that $y=f(x)$ .

Let $M$ be the fiber of the map $f$ at the point $y$ , i.e. $M=f^{-1}(y)$ . By assumption, $M$ is countably tight. Call a point $w \in M$ countably reached by $A$ if there is some countable $C \subset A$ such that $w \in \overline{C}$ . Let $G$ be the set of all points in $M$ that are countably reached by $A$ .

We claim that $x \in \overline{G}$ . Let $U \subset Y_1$ be open such that $x \in U$ . Because the space $Y_1$ is regular, choose open $V \subset U$ such that $x \in V$ and $\overline{V} \subset U$ . Then $V \cap A \ne \varnothing$ . Furthermore, $x \in \overline{V \cap A}$ . Let $C=f(V \cap A)$ . According to the fact about continuity stated above, we have $y \in \overline{C}$ . Since $Y_2$ is countably tight, there exists some countable $D \subset C$ such that $y \in \overline{D}$ . Choose a countable $E \subset V \cap A$ such that $f(E)=D$ . It follows that $y \in \overline{f(E)}$ .

We show that that $\overline{E} \cap M \ne \varnothing$ . Since $E \subset \overline{E}$ , we have $f(E) \subset f(\overline{E})$ . Note that $f(\overline{E})$ is a closed set since $f$ is a closed map. Thus $\overline{f(E)} \subset f(\overline{E})$ . As a result, $y \in f(\overline{E})$ . Then $y=f(t)$ for some $t \in \overline{E}$ . We have $t \in \overline{E} \cap M$ .

By the definition of the set $G$ , we have $\overline{E} \cap M \subset G$ . With $E \subset V \cap A$ , we have $\overline{E} \subset \overline{V \cap A} \subset \overline{V}$ . Thus, $\overline{E} \cap M \subset \overline{V} \subset U$ . Note that the arbitrary open neighborhood $U$ of $x$ contains points of $G$ . This establishes the claim that $x \in \overline{G}$ .

Since $M$ is a fiber of $f$ , $M$ is countably tight by assumption. Since $G \subset M$ , $G$ is also countably tight. Since $x \in M$ and $x \in \overline{G}$ , we can choose some countable $T \subset G$ such that $x \in \overline{T}$ . For each $t \in T$ , choose a countable $W_t \subset A$ with $t \in \overline{W_t}$ . Let $W=\bigcup_{t \in T} W_t$ . Note that $W \subset A$ and $W$ is countable with $x \in \overline{W}$ . This establishes the space $Y_1$ is countably tight at $x \in Y_1$ . $\blacksquare$

Lemma 5
Let $f:X \times Y \rightarrow Y$ be the projection map. If $X$ is a compact space, then $f$ is a closed map.

Proof of Lemma 5
Let $A$ be a closed subset of $X \times Y$ . Suppose that $f(A)$ is not closed. Let $y \in \overline{f(A)}-f(A)$ . It follows that no point of $X \times \left\{y \right\}$ belongs to $A$ . For each $x \in X$ , choose open subset $O_x$ of $X \times Y$ such that $(x,y) \in O_x$ and $O_x \cap A=\varnothing$ . The set of all $O_x$ is an open cover of the compact space $X \times \left\{y \right\}$ . Then there exist finitely many $O_x$ that cover $X \times \left\{y \right\}$ , say $O_{x_i}$ for $i=1,2,\cdots,n$ .

Let $W=\bigcup_{i=1}^n O_{x_i}$ . We have $X \times \left\{y \right\} \subset W$ . Since $X$ is compact, we can then use the Tube Lemma which implies that there exists open $G \subset Y$ such that $X \times \left\{y \right\} \subset X \times G \subset W$ . It follows that $G \cap f(A) \ne \varnothing$ . Choose $t \in G \cap f(A)$ . Then for some $x \in X$ , $(x,t) \in A$ . Since $t \in G$ , $(x,t) \in W$ , implying that $W \cap A \ne \varnothing$ , a contradiction. Thus $f(A)$ must be a closed set in $Y$ . This completes the proof of the lemma. $\blacksquare$

Proof of Theorem 1
Let $X$ be the factor that is compact. Let $f: X \times Y \rightarrow Y$ be the projection map. The projection map is always continuous. Furthermore it is a closed map by Lemma 5. The range space $Y$ is countably tight by assumption. Each fiber of the projection map $f$ is of the form $X \times \left\{y \right\}$ where $y \in Y$ , which is countably tight. Then use Theorem 4 to establish that $X \times Y$ is countably tight. $\blacksquare$

Proof of Theorem 2
This is a corollary of Theorem 1. According to Theorem 1, the product of two compact countably tight spaces is countably tight. By induction, the product of any finite number of compact countably tight spaces is countably tight. $\blacksquare$

Countable products

Our proof to establish that the product space $\prod_{i=1}^\infty X_i$ is countably tight is an indirect one and makes use of two non-trivial results. We first show that $\omega_1 \times \prod_{i=1}^\infty X_i$ is a closed subspace of a $\Sigma$ -product that is normal. It follows from another result that the second factor $\prod_{i=1}^\infty X_i$ is countably tight. We now present all the necessary definitions and theorems.

Consider a product space $Y=\prod_{\alpha<\kappa} Y_\alpha$ where $\kappa$ is an infinite cardinal number. Fix a point $p \in Y$ . The $\Sigma$ -product of the spaces $Y_\alpha$ with $p$ as the base point is the following subspace of the product space $Y=\prod_{\alpha<\kappa} Y_\alpha$ :

$\displaystyle \Sigma_{\alpha<\kappa} Y_\alpha=\left\{y \in \prod_{\alpha<\kappa} Y_\alpha: y_\alpha \ne p_\alpha \text{ for at most countably many } \alpha < \kappa \right\}$

The definition of the space $\Sigma_{\alpha<\kappa} Y_\alpha$ depends on the base point $p$ . The discussion here is on properties of $\Sigma_{\alpha<\kappa} Y_\alpha$ that hold regardless of the choice of base point. If the factor spaces are indexed by a set $A$ , the notation is $\Sigma_{\alpha \in A} Y_\alpha$ .

If all factors $Y_\alpha$ are identical, say $Y_\alpha=Z$ for all $\alpha$ , then we use the notation $\Sigma_{\alpha<\kappa} Z$ to denote the $\Sigma$ -product. Once useful fact is that if there are $\omega_1$ many factors and each factor has at least 2 points, then the space $\omega_1$ can be embedded as a closed subspace of the $\Sigma$ -product.

Theorem 6
For each $\alpha<\omega_1$ , let $Y_\alpha$ be a space with at least two points. Then $\Sigma_{\alpha<\omega_1} Y_\alpha$ contains $\omega_1$ as a closed subspace. See Exercise 3 in this previous post.

Now we discuss normality of $\Sigma$ -products. This previous post shows that if each factor is a separable metric space, then the $\Sigma$ -product is normal. It is also well known that if each factor is a metric space, the $\Sigma$ -product is normal. The following theorem handles the case where each factor is a compact space.

Theorem 7
For each $\alpha<\kappa$ , let $Y_\alpha$ be a compact space. Then the $\Sigma$ -product $\Sigma_{\alpha<\kappa} Y_\alpha$ is normal if and only if each factor $Y_\alpha$ is countably tight.

Theorem 7 is Theorem 7.5 in page 821 of [1]. Theorem 7.5 in [1] is stated in a more general setting where each factor of the $\Sigma$ -product is a paracompact p-space. We will not go into a discussion of p-space. It suffices to know that any compact Hausdorff space is a paracompact p-space. We also need the following theorem, which is proved in this previous post.

Theorem 8
Let $Y$ be a compact space. Then the product space $\omega_1 \times Y$ is normal if and only if $Y$ is countably tight.

We now prove Theorem 3.

Proof of Theorem 3
Let $\omega_1=\cup \left\{A_n: n \in \omega \right\}$ , where for each $n$ , $\lvert A_n \lvert=\omega_1$ and that $A_n \cap A_m=\varnothing$ if $n \ne m$ . For each $n=1,2,3,\cdots$ , let $S_n=\Sigma_{\alpha \in A_n} X_n$ . By Theorem 7, each $S_n$ is normal. Let $S_0=\Sigma_{\alpha \in A_0} X_1$ , which is also normal. By Theorem 6, the space $\omega_1$ of countable ordinals is a closed subspace of $S_0$ . Let $T=\omega_1 \times X_1 \times X_2 \times X_3 \times \cdots$ . We have the following derivation.

$\displaystyle \begin{aligned} T&=\omega_1 \times X_1 \times X_2 \times X_3 \times \cdots \\&\subset S_0 \times S_1 \times S_2 \times S_3 \times \cdots \\&\cong W=\Sigma_{\alpha<\omega_1} W_\alpha \end{aligned}$

Recall that $\omega_1=\cup \left\{A_n: n \in \omega \right\}$ . The space $W=\Sigma_{\alpha<\omega_1} W_\alpha$ is defined such that for each $n \ge 1$ and for each $\alpha \in A_n$ , $W_\alpha=X_n$ . Furthermore, for $n=0$ , for each $\alpha \in A_0$ , let $W_\alpha=X_1$ . Thus $W$ is a $\Sigma$ -product of compact countably tight spaces and is thus normal by Theorem 7. The space $T=\omega_1 \times \prod_{n=1}^\infty X_n$ is a closed subspace of the normal space $W$ . By Theorem 8, the product space $\prod_{n=1}^\infty X_n$ must be countably tight. $\blacksquare$

Remarks

Theorem 2, as indicated above, is a corollary of Theorem 1. We also note that Theorem 2 is also a corollary of Theorem 3 since any finite product is a subspace of a countable product. To see this, let $X=X_1 \times X_2 \times \cdots \times X_n$ .

$\displaystyle \begin{aligned} X&=X_1 \times X_2 \times \cdots \times X_n \\&\cong X_1 \times X_2 \times \cdots \times X_n \times \left\{t_{n+1} \right\} \times \left\{t_{n+2} \right\} \times \cdots \\&\subset X_1 \times X_2 \times \cdots \times X_n \times X_{n+1} \times X_{n+2} \times \cdots \end{aligned}$

In the above derivation, $t_m$ is a point of $X_m$ for all $m >n$ . When the countable product space is countably tight, the finite product, being a subspace of a countably tight space, is also countably tight.

Exercise

Exercise 1
Let $f:X \times Y \rightarrow Y$ be the projection map. If $X$ is a countably compact space and $Y$ is a Frechet space, then $f$ is a closed map.

Exercise 2
Let $X$ and $Y$ be countably tight spaces. If one of $X$ and $Y$ is a countably compact space and the other space is a Frechet space, then $X \times Y$ is countably tight.

Exercise 2 is a variation of Theorem 1. One factor is weakened to “countably compact”. However, the other factor is strengthened to “Frechet”.

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.

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Posted: July 1, 2015
Revised: April 17, 2023

Normality in the powers of countably compact spaces

Posted on May 17, 2015 by Dan Ma

Let $\omega_1$ be the first uncountable ordinal. The topology on $\omega_1$ we are interested in is the ordered topology, the topology induced by the well ordering. The space $\omega_1$ is also called the space of all countable ordinals since it consists of all ordinals that are countable in cardinality. It is a handy example of a countably compact space that is not compact. In this post, we consider normality in the powers of $\omega_1$ . We also make comments on normality in the powers of a countably compact non-compact space.

Let $\omega$ be the first infinite ordinal. It is well known that $\omega^{\omega_1}$ , the product space of $\omega_1$ many copies of $\omega$ , is not normal (a proof can be found in this earlier post). This means that any product space $\prod_{\alpha<\kappa} X_\alpha$ , with uncountably many factors, is not normal as long as each factor $X_\alpha$ contains a countable discrete space as a closed subspace. Thus in order to discuss normality in the product space $\prod_{\alpha<\kappa} X_\alpha$ , the interesting case is when each factor is infinite but contains no countable closed discrete subspace (i.e. no closed copies of $\omega$ ). In other words, the interesting case is that each factor $X_\alpha$ is a countably compact space that is not compact (see this earlier post for a discussion of countably compactness). In particular, we would like to discuss normality in $X^{\kappa}$ where $X$ is a countably non-compact space. In this post we start with the space $X=\omega_1$ of the countable ordinals. We examine $\omega_1$ power $\omega_1^{\omega_1}$ as well as the countable power $\omega_1^{\omega}$ . The former is not normal while the latter is normal. The proof that $\omega_1^{\omega}$ is normal is an application of the normality of $\Sigma$ -product of the real line.

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The uncountable product

Theorem 1
The product space $\prod_{\alpha<\omega_1} \omega_1=\omega_1^{\omega_1}$ is not normal.

Theorem 1 follows from Theorem 2 below. For any space $X$ , a collection $\mathcal{C}$ of subsets of $X$ is said to have the finite intersection property if for any finite $\mathcal{F} \subset \mathcal{C}$ , the intersection $\cap \mathcal{F} \ne \varnothing$ . Such a collection $\mathcal{C}$ is called an f.i.p collection for short. It is well known that a space $X$ is compact if and only collection $\mathcal{C}$ of closed subsets of $X$ satisfying the finite intersection property has non-empty intersection (see Theorem 1 in this earlier post). Thus any non-compact space has an f.i.p. collection of closed sets that have empty intersection.

In the space $X=\omega_1$ , there is an f.i.p. collection of cardinality $\omega_1$ using its linear order. For each $\alpha<\omega_1$ , let $C_\alpha=\left\{\beta<\omega_1: \alpha \le \beta \right\}$ . Let $\mathcal{C}=\left\{C_\alpha: \alpha < \omega_1 \right\}$ . It is a collection of closed subsets of $X=\omega_1$ . It is an f.i.p. collection and has empty intersection. It turns out that for any countably compact space $X$ with an f.i.p. collection of cardinality $\omega_1$ that has empty intersection, the product space $X^{\omega_1}$ is not normal.

Theorem 2
Let $X$ be a countably compact space. Suppose that there exists a collection $\mathcal{C}=\left\{C_\alpha: \alpha < \omega_1 \right\}$ of closed subsets of $X$ such that $\mathcal{C}$ has the finite intersection property and that $\mathcal{C}$ has empty intersection. Then the product space $X^{\omega_1}$ is not normal.

Proof of Theorem 2
Let’s set up some notations on product space that will make the argument easier to follow. By a standard basic open set in the product space $X^{\omega_1}=\prod_{\alpha<\omega_1} X$ , we mean a set of the form $O=\prod_{\alpha<\omega_1} O_\alpha$ such that each $O_\alpha$ is an open subset of $X$ and that $O_\alpha=X$ for all but finitely many $\alpha<\omega_1$ . Given a standard basic open set $O=\prod_{\alpha<\omega_1} O_\alpha$ , the notation $\text{Supp}(O)$ refers to the finite set of $\alpha$ for which $O_\alpha \ne X$ . For any set $M \subset \omega_1$ , the notation $\pi_M$ refers to the projection map from $\prod_{\alpha<\omega_1} X$ to the subproduct $\prod_{\alpha \in M} X$ . Each element $d \in X^{\omega_1}$ can be considered a function $d: \omega_1 \rightarrow X$ . By $(d)_\alpha$ , we mean $(d)_\alpha=d(\alpha)$ .

For each $t \in X$ , let $f_t: \omega_1 \rightarrow X$ be the constant function whose constant value is $t$ . Consider the following subspaces of $X^{\omega_1}$ .

$H=\prod_{\alpha<\omega_1} C_\alpha$

$\displaystyle K=\left\{f_t: t \in X \right\}$

Both $H$ and $K$ are closed subsets of the product space $X^{\omega_1}$ . Because the collection $\mathcal{C}$ has empty intersection, $H \cap K=\varnothing$ . We show that $H$ and $K$ cannot be separated by disjoint open sets. To this end, let $U$ and $V$ be open subsets of $X^{\omega_1}$ such that $H \subset U$ and $K \subset V$ .

Let $d_1 \in H$ . Choose a standard basic open set $O_1$ such that $d_1 \in O_1 \subset U$ . Let $S_1=\text{Supp}(O_1)$ . Since $S_1$ is the support of $O_1$ , it follows that $\pi_{S_1}^{-1}(\pi_{S_1}(d_1)) \subset O_1 \subset U$ . Since $\mathcal{C}$ has the finite intersection property, there exists $a_1 \in \bigcap_{\alpha \in S_1} C_\alpha$ .

Define $d_2 \in H$ such that $(d_2)_\alpha=a_1$ for all $\alpha \in S_1$ and $(d_2)_\alpha=(d_1)_\alpha$ for all $\alpha \in \omega_1-S_1$ . Choose a standard basic open set $O_2$ such that $d_2 \in O_2 \subset U$ . Let $S_2=\text{Supp}(O_2)$ . It is possible to ensure that $S_1 \subset S_2$ by making more factors of $O_2$ different from $X$ . We have $\pi_{S_2}^{-1}(\pi_{S_2}(d_2)) \subset O_2 \subset U$ . Since $\mathcal{C}$ has the finite intersection property, there exists $a_2 \in \bigcap_{\alpha \in S_2} C_\alpha$ .

Now choose a point $d_3 \in H$ such that $(d_3)_\alpha=a_2$ for all $\alpha \in S_2$ and $(d_3)_\alpha=(d_2)_\alpha$ for all $\alpha \in \omega_1-S_2$ . Continue on with this inductive process. When the inductive process is completed, we have the following sequences:

a sequence $d_1,d_2,d_3,\cdots$ of point of $H=\prod_{\alpha<\omega_1} C_\alpha$ ,
a sequence $S_1 \subset S_2 \subset S_3 \subset \cdots$ of finite subsets of $\omega_1$ ,
a sequence $a_1,a_2,a_3,\cdots$ of points of $X$

such that for all $n \ge 2$ , $(d_n)_\alpha=a_{n-1}$ for all $\alpha \in S_{n-1}$ and $\pi_{S_n}^{-1}(\pi_{S_n}(d_n)) \subset U$ . Let $A=\left\{a_1,a_2,a_3,\cdots \right\}$ . Either $A$ is finite or $A$ is infinite. Let’s examine the two cases.

Case 1
Suppose that $A$ is infinite. Since $X$ is countably compact, $A$ has a limit point $a$ . That means that every open set containing $a$ contains some $a_n \ne a$ . For each $n \ge 2$ , define $y_n \in \prod_{\alpha< \omega_1} X$ such that

$(y_n)_\alpha=(d_n)_\alpha=a_{n-1}$ for all $\alpha \in S_n$ ,
$(y_n)_\alpha=a$ for all $\alpha \in \omega_1-S_n$

From the induction step, we have $y_n \in \pi_{S_n}^{-1}(\pi_{S_n}(d_n)) \subset U$ for all $n$ . Let $t=f_a \in K$ , the constant function whose constant value is $a$ . It follows that $t$ is a limit of $\left\{y_1,y_2,y_3,\cdots \right\}$ . This means that $t \in \overline{U}$ . Since $t \in K \subset V$ , $U \cap V \ne \varnothing$ .

Case 2
Suppose that $A$ is finite. Then there is some $m$ such that $a_m=a_j$ for all $j \ge m$ . For each $n \ge 2$ , define $y_n \in \prod_{\alpha< \omega_1} X$ such that

$(y_n)_\alpha=(d_n)_\alpha=a_{n-1}$ for all $\alpha \in S_n$ ,
$(y_n)_\alpha=a_m$ for all $\alpha \in \omega_1-S_n$

As in Case 1, we have $y_n \in \pi_{S_n}^{-1}(\pi_{S_n}(d_n)) \subset U$ for all $n$ . Let $t=f_{a_m} \in K$ , the constant function whose constant value is $a_m$ . It follows that $t=y_n$ for all $n \ge m+1$ . Thus $U \cap V \ne \varnothing$ .

Both cases show that $U \cap V \ne \varnothing$ . This completes the proof the product space $X^{\omega_1}$ is not normal. $\blacksquare$

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The countable product

Theorem 3
The product space $\prod_{\alpha<\omega} \omega_1=\omega_1^{\omega}$ is normal.

Proof of Theorem 3
The proof here actually proves more than normality. It shows that $\prod_{\alpha<\omega} \omega_1=\omega_1^{\omega}$ is collectionwise normal, which is stronger than normality. The proof makes use of the $\Sigma$ -product of $\kappa$ many copies of $\mathbb{R}$ , which is the following subspace of the product space $\mathbb{R}^{\kappa}$ .

$\Sigma(\kappa)=\left\{x \in \mathbb{R}^{\kappa}: x(\alpha) \ne 0 \text{ for at most countably many } \alpha<\kappa \right\}$

It is well known that $\Sigma(\kappa)$ is collectionwise normal (see this earlier post). We show that $\prod_{\alpha<\omega} \omega_1=\omega_1^{\omega}$ is a closed subspace of $\Sigma(\kappa)$ where $\kappa=\omega_1$ . Thus $\omega_1^{\omega}$ is collectionwise normal. This is established in the following claims.

Claim 1
We show that the space $\omega_1$ is embedded as a closed subspace of $\Sigma(\omega_1)$ .

For each $\beta<\omega_1$ , define $f_\beta:\omega_1 \rightarrow \mathbb{R}$ such that $f_\beta(\gamma)=1$ for all $\gamma<\beta$ and $f_\beta(\gamma)=0$ for all $\beta \le \gamma <\omega_1$ . Let $W=\left\{f_\beta: \beta<\omega_1 \right\}$ . We show that $W$ is a closed subset of $\Sigma(\omega_1)$ and $W$ is homeomorphic to $\omega_1$ according to the mapping $f_\beta \rightarrow W$ .

First, we show $W$ is closed by showing that $\Sigma(\omega_1)-W$ is open. Let $y \in \Sigma(\omega_1)-W$ . We show that there is an open set containing $y$ that contains no points of $W$ .

Suppose that for some $\gamma<\omega_1$ , $y_\gamma \in O=\mathbb{R}-\left\{0,1 \right\}$ . Consider the open set $Q=(\prod_{\alpha<\omega_1} Q_\alpha) \cap \Sigma(\omega_1)$ where $Q_\alpha=\mathbb{R}$ except that $Q_\gamma=O$ . Then $y \in Q$ and $Q \cap W=\varnothing$ .

So we can assume that for all $\gamma<\omega_1$ , $y_\gamma \in \left\{0, 1 \right\}$ . There must be some $\theta$ such that $y_\theta=1$ . Otherwise, $y=f_0 \in W$ . Since $y \ne f_\theta$ , there must be some $\delta<\gamma$ such that $y_\delta=0$ . Now choose the open interval $T_\theta=(0.9,1.1)$ and the open interval $T_\delta=(-0.1,0.1)$ . Consider the open set $M=(\prod_{\alpha<\omega_1} M_\alpha) \cap \Sigma(\omega_1)$ such that $M_\alpha=\mathbb{R}$ except for $M_\theta=T_\theta$ and $M_\delta=T_\delta$ . Then $y \in M$ and $M \cap W=\varnothing$ . We have just established that $W$ is closed in $\Sigma(\omega_1)$ .

Consider the mapping $f_\beta \rightarrow W$ . Based on how it is defined, it is straightforward to show that it is a homeomorphism between $\omega_1$ and $W$ .

Claim 2
The $\Sigma$ -product $\Sigma(\omega_1)$ has the interesting property it is homeomorphic to its countable power, i.e.

$\Sigma(\omega_1) \cong \Sigma(\omega_1) \times \Sigma(\omega_1) \times \Sigma(\omega_1) \cdots \ \ \ \ \ \ \ \ \ \ \ \text{(countably many times)}$

Because each element of $\Sigma(\omega_1)$ is nonzero only at countably many coordinates, concatenating countably many elements of $\Sigma(\omega_1)$ produces an element of $\Sigma(\omega_1)$ . Thus Claim 2 can be easily verified. With above claims, we can see that

$\displaystyle \omega_1^{\omega}=\omega_1 \times \omega_1 \times \omega_1 \times \cdots \subset \Sigma(\omega_1) \times \Sigma(\omega_1) \times \Sigma(\omega_1) \cdots \cong \Sigma(\omega_1)$

Thus $\omega_1^{\omega}$ is a closed subspace of $\Sigma(\omega_1)$ . Any closed subspace of a collectionwise normal space is collectionwise normal. We have established that $\omega_1^{\omega}$ is normal. $\blacksquare$

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The normality in the powers of $X$

We have established that $\prod_{\alpha<\omega_1} \omega_1=\omega_1^{\omega_1}$ is not normal. Hence any higher uncountable power of $\omega_1$ is not normal. We have also established that $\prod_{\alpha<\omega} \omega_1=\omega_1^{\omega}$ , the countable power of $\omega_1$ is normal (in fact collectionwise normal). Hence any finite power of $\omega_1$ is normal. However $\omega_1^{\omega}$ is not hereditarily normal. One of the exercises below is to show that $\omega_1 \times \omega_1$ is not hereditarily normal.

Theorem 2 can be generalized as follows:

Theorem 4
Let $X$ be a countably compact space has an f.i.p. collection $\mathcal{C}$ of closed sets such that $\bigcap \mathcal{C}=\varnothing$ . Then $X^{\kappa}$ is not normal where $\kappa=\lvert \mathcal{C} \lvert$ .

The proof of Theorem 2 would go exactly like that of Theorem 2. Consider the following two theorems.

Theorem 5
Let $X$ be a countably compact space that is not compact. Then there exists a cardinal number $\kappa$ such that $X^{\kappa}$ is not normal and $X^{\tau}$ is normal for all cardinal number $\tau<\kappa$ .

By the non-compactness of $X$ , there exists an f.i.p. collection $\mathcal{C}$ of closed subsets of $X$ such that $\bigcap \mathcal{C}=\varnothing$ . Let $\kappa$ be the least cardinality of such an f.i.p. collection. By Theorem 4, that $X^{\kappa}$ is not normal. Because $\kappa$ is least, any smaller power of $X$ must be normal.

Theorem 6
Let $X$ be a space that is not countably compact. Then $X^{\kappa}$ is not normal for any cardinal number $\kappa \ge \omega_1$ .

Since the space $X$ in Theorem 6 is not countably compact, it would contain a closed and discrete subspace that is countable. By a theorem of A. H. Stone, $\omega^{\omega_1}$ is not normal. Then $\omega^{\omega_1}$ is a closed subspace of $X^{\omega_1}$ .

Thus between Theorem 5 and Theorem 6, we can say that for any non-compact space $X$ , $X^{\kappa}$ is not normal for some cardinal number $\kappa$ . The $\kappa$ from either Theorem 5 or Theorem 6 is at least $\omega_1$ . Interestingly for some spaces, the $\kappa$ can be much smaller. For example, for the Sorgenfrey line, $\kappa=2$ . For some spaces (e.g. the Michael line), $\kappa=\omega$ .

Theorems 4, 5 and 6 are related to a theorem that is due to Noble.

Theorem 7 (Noble)
If each power of a space $X$ is normal, then $X$ is compact.

A proof of Noble’s theorem is given in this earlier post, the proof of which is very similar to the proof of Theorem 2 given above. So the above discussion the normality of powers of $X$ is just another way of discussing Theorem 7. According to Theorem 7, if $X$ is not compact, some power of $X$ is not normal.

The material discussed in this post is excellent training ground for topology. Regarding powers of countably compact space and product of countably compact spaces, there are many topics for further discussion/investigation. One possibility is to examine normality in $X^{\kappa}$ for more examples of countably compact non-compact $X$ . One particular interesting example would be a countably compact non-compact $X$ such that the least power $\kappa$ for non-normality in $X^{\kappa}$ is more than $\omega_1$ . A possible candidate could be the second uncountable ordinal $\omega_2$ . By Theorem 2, $\omega_2^{\omega_2}$ is not normal. The issue is whether the $\omega_1$ power $\omega_2^{\omega_1}$ and countable power $\omega_2^{\omega}$ are normal.

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Exercises

Exercise 1
Show that $\omega_1 \times \omega_1$ is not hereditarily normal.

Exercise 2
Show that the mapping $f_\beta \rightarrow W$ in Claim 3 in the proof of Theorem 3 is a homeomorphism.

Exercise 3
The proof of Theorem 3 shows that the space $\omega_1$ is a closed subspace of the $\Sigma$ -product of the real line. Show that $\omega_1$ can be embedded in the $\Sigma$ -product of arbitrary spaces.

For each $\alpha<\omega_1$ , let $X_\alpha$ be a space with at least two points. Let $p \in \prod_{\alpha<\omega_1} X_\alpha$ . The $\Sigma$ -product of the spaces $X_\alpha$ is the following subspace of the product space $\prod_{\alpha<\omega_1} X_\alpha$ .

$\Sigma(X_\alpha)=\left\{x \in \prod_{\alpha<\omega_1} X_\alpha: x(\alpha) \ne p(\alpha) \ \text{for at most countably many } \alpha<\omega_1 \right\}$

The point $p$ is the center of the $\Sigma$ -product. Show that the space $\Sigma(X_\alpha)$ contains $\omega_1$ as a closed subspace.

Exercise 4
Find a direct proof of Theorem 3, that $\omega_1^{\omega}$ is normal.

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$\copyright \ 2015 \text{ by Dan Ma}$

The product of uncountably many factors is never hereditarily normal

Posted on May 3, 2015 by Dan Ma

The space $Y=\prod_{\alpha<\omega_1} \left\{0,1 \right\}=\left\{0,1 \right\}^{\omega_1}$ is the product of $\omega_1$ many copies of the two-element set $\left\{0,1 \right\}$ where $\omega_1$ is the first uncountable ordinal. It is a compact space by Tychonoff’s theorem. It is a normal space since every compact Hausdorff space is normal. A space is hereditarily normal if every subspace is normal. Is the space $Y$ hereditarily normal? In this post, we give two proofs that it is not hereditarily normal. It then follows that any product space $\prod X_\alpha$ cannot be hereditarily normal as long as there are uncountably many factors and every factor has at least two point.

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The connection with a theorem of Katetov

It turns out that there is a connection with a theorem of Katetov. For any compact space, knowing hereditary normality of the first several self product spaces can reveal a great deal of information about the compact space. More specifically, for any compact space $X$ , knowing whether $X$ , $X^2$ and $X^3$ are hereditarily normal can tell us whether $X$ is metrizable. If all three are hereditarily normal, then $X$ is metrizable. If one of the three self products is not hereditarily normal, then $X$ is not metrizable. This fact is based on a theorem of Katetov (see this previous post). The space $Y=\left\{0,1 \right\}^{\omega_1}$ is not metrizable since it is not first countable (see Problem 1 below). Thus one of its first three self products must fail to be hereditarily normal.

These two proofs are not direct proof in the sense that a non-normal subspace is not explicitly produced. Instead the proofs use other theorem or basic but important background results. One of the two proofs (#2) uses a theorem of Katetov on hereditarily normal spaces. The other proof (#1) uses the fact that the product of uncountably many copies of a countable discrete space is not normal. We believe that these two proofs and the required basic facts are an important training ground for topology. We list out these basic facts as exercises. Anyone who wishes to fill in the gaps can do so either by studying the links provided or by consulting other sources.

The theorem of Katetov mentioned earlier provides a great exercise – for any non-metrizable compact space $X$ , determine where the hereditary normality fails. Does it fail in $X$ , $X^2$ or $X^3$ ? This previous post examines a small list of compact non-metrizable spaces. In all the examples in this list, the hereditary normality fails in $X$ or $X^2$ . The space $Y=\left\{0,1 \right\}^{\omega_1}$ can be added to this list. All the examples in this list are defined using no additional set theory axioms beyond ZFC. A natural question: does there exist an example of compact non-metrizable space $X$ such that the hereditary normality holds in $X^2$ and fails in $X^3$ ? It turns out that this was a hard problem and the answer is independent of ZFC. This previous post provides a brief discussion and has references for the problem.

All spaces under consideration are Hausdorff spaces.

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Exercises

Problem 1
Let $X$ be a compact space. Show that $X$ is normal.

Problem 2
For each $\alpha<\omega_1$ , let $A_\alpha$ be a set with cardinality $\le \omega_1$ . Show that $\lvert \bigcup_{\alpha<\omega_1} A_\alpha \lvert \le \omega_1$ .

Problem 2 holds for any infinite cardinal, not just $\omega_1$ . One reference for Problem 2 is Lemma 10.21 on page 30 of Set Theorey, An Introduction to Independence Proofs by Kenneth Kunen.

Problem 3
For each $\alpha<\omega_1$ , let $X_\alpha$ be a space with at least two points. Show that for every point $p \in \prod_{\alpha<\omega_1} X_\alpha$ , there does not exist a countable base at the point $p$ . In other words, the product space $\prod_{\alpha<\omega_1} X_\alpha$ is not first countable at every point. It follows that product space $\prod_{\alpha<\omega_1} X_\alpha$ is not metrizable.

Problem 4
In any space, a $G_\delta$ -set is a set that is the intersection of countably many open sets. When a singleton set $\left\{ x \right\}$ is a $G_\delta$ -set, we say the point $x$ is a $G_\delta$ -point. For each $\alpha<\omega_1$ , let $X_\alpha$ be a space with at least two points. Show that every point $p$ in the product space $\prod_{\alpha<\omega_1} X_\alpha$ is not a $G_\delta$ -point.

Note that Problem 4 implies Problem 3.

For Problem 3 and Problem 4, use the fact that there are uncountably many factors and that a basic open set in the product space is of the form $\prod_{\alpha<\omega_1} O_\alpha$ and that it has only finitely many coordinates at which $O_\alpha \ne X_\alpha$ .

Problem 5
For each $\alpha<\omega_1$ , let $X_\alpha=\left\{0,1,2,\cdots \right\}$ be the set of non-negative integers with the discrete topology. Show that the product space $\prod_{\alpha<\omega_1} X_\alpha$ is not normal.

See here for a discussion of Problem 5.

Problem 6
Let $\displaystyle Y=\left\{0,1 \right\}^{\omega_1}$ . Show that $Y$ has a countably infinite subspace

$W=\left\{y_0,y_1,y_2,y_3\cdots \right\}$

such that $W$ is relatively discrete. In other words, $W$ is discrete in the subspace topology of $W$ . However $W$ is not discrete in the product space $Y$ since $Y$ is compact.

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Proof #1

Let $\displaystyle Y=\left\{0,1 \right\}^{\omega_1}$ . We show that $Y$ is not hereditarily normal.

Note that the product space $\displaystyle Y=\left\{0,1 \right\}^{\omega_1}$ can be written as the product of $\omega_1$ many copies of itself:

$\displaystyle \left\{0,1 \right\}^{\omega_1} \cong \left\{0,1 \right\}^{\omega_1} \times \left\{0,1 \right\}^{\omega_1} \times \left\{0,1 \right\}^{\omega_1} \times \cdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

The fact (1) follows from the fact that the union of $\omega_1$ many pairwise disjoint sets, each of which has cardinality $\omega_1$ , has cardinality $\omega_1$ (see Problem 2). The space $\left\{0,1 \right\}^{\omega_1}$ has a countably infinite subspace that is relatively discrete (see Problem 6). In other words, it has a subspace that is homemorphic to $\omega=\left\{0,1,2,\cdots \right\}$ where $\omega$ has the discrete topology. Thus the following is homeomorphic to a subspace of $\displaystyle Y=\left\{0,1 \right\}^{\omega_1}$ .

$\displaystyle \omega^{\omega_1} = \omega \times \omega \times \omega \times \cdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

By Problem 5, the space $\omega^{\omega_1}$ is not normal. Hence the compact space $\displaystyle Y=\left\{0,1 \right\}^{\omega_1}$ contains the non-normal space $\omega^{\omega_1}$ and is thus not hereditarily normal. $\blacksquare$

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Proof #2

Let $\displaystyle Y=\left\{0,1 \right\}^{\omega_1}$ . We show that $Y$ is not hereditarily normal. This proof uses a theorem of Katetov, discussed in this previous post and stated below.

Theorem 1
If $X_1 \times X_2$ is hereditarily normal (i.e. every one of its subspaces is normal), then one of the following condition holds:

The factor $X_1$ is perfectly normal.
Every countable and infinite subset of the factor $X_2$ is closed.

First, $Y$ can be written as the product of two copies of itself:

$\displaystyle \left\{0,1 \right\}^{\omega_1} \cong \left\{0,1 \right\}^{\omega_1} \times \left\{0,1 \right\}^{\omega_1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$

This is because the union of two disjoints sets, each of which has cardinality $\omega_1$ , has carinality $\omega_1$ . Note that the countably infinite subset $W$ from Problem 6 is not a closed subset of $Y$ . If it were, the compact space $Y$ would contain an infinite set with no limit point. Thus the second condition of Theorem 1 is not satisfied. If $Y \cong Y \times Y$ were to be hereditarily normal, then the first condition must be satisfied, i.e. $Y$ is perfectly normal (meaning that $Y$ is normal and that every closed subset of it is a $G_\delta$ -set). However, Problem 4 indicates that no point in $Y$ can be a $G_\delta$ point. Therefore $Y$ cannot be hereditarily normal. $\blacksquare$

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Corollary

The product of uncountably many spaces, each one of which has at least two points, contains a homeomorphic copy of the space $\displaystyle Y=\left\{0,1 \right\}^{\omega_1}$ . Thus such a product space can never be hereditarily normal. We state this more formally below.

Theorem 2
Let $\kappa$ be any uncountable cardinal. For each $\alpha<\kappa$ , let $X_\alpha$ be a space with at least two points. Then $\prod_{\alpha<\kappa} X_\alpha$ is not hereditarily normal.

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$\copyright \ 2015 \text{ by Dan Ma}$

Looking for non-normal subspaces of the square of a compact X

Posted on April 14, 2015 by Dan Ma

A theorem of Katetov states that if $X$ is compact with a hereditarily normal cube $X^3$ , then $X$ is metrizable (discussed in this previous post). This means that for any non-metrizable compact space $X$ , Katetov’s theorem guarantees that some subspace of the cube $X^3$ is not normal. Where can a non-normal subspace of $X^3$ be found? Is it in $X$ , in $X^2$ or in $X^3$ ? In other words, what is the “dimension” in which the hereditary normality fails for a given compact non-metrizable $X$ (1, 2 or 3)? Katetov’s theorem guarantees that the dimension must be at most 3. Out of curiosity, we gather a few compact non-metrizable spaces. They are discussed below. In this post, we motivate an independence result using these examples.

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Katetov’s theorems

First we state the results of Katetov for reference. These results are proved in this previous post.

Theorem 1
If $X \times Y$ is hereditarily normal (i.e. every one of its subspaces is normal), then one of the following condition holds:

The factor $X$ is perfectly normal.
Every countable and infinite subset of the factor $Y$ is closed.

Theorem 2
If $X$ and $Y$ are compact and $X \times Y$ is hereditarily normal, then both $X$ and $Y$ are perfectly normal.

Theorem 3
Let $X$ be a compact space. If $X^3=X \times X \times X$ is hereditarily normal, then $X$ is metrizable.

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Examples of compact non-metrizable spaces

The set-theoretic result presented here is usually motivated by looking at Theorem 3. The question is: Can $X^3$ in Theorem 3 be replaced by $X^2$ ? We take a different angle of looking at some standard compact non-metric spaces and arrive at the same result. The following is a small listing of compact non-metrizable spaces. Each example in this list is defined in ZFC alone, i.e. no additional axioms are used beyond the generally accepted axioms of set theory.

One-point compactification of the Tychonoff plank.
One-point compactification of $\psi(\mathcal{A})$ where $\mathcal{A}$ is a maximal almost disjoint family of subsets of $\omega$ .
The first compact uncountable ordinal, i.e. $\omega_1+1$ .
The one-point compactification of an uncountable discrete space.
Alexandroff double circle.
Double arrow space.
Unit square with the lexicographic order.

Since each example in the list is compact and non-metrizable, the cube of each space must not be hereditarily normal according to Theorem 3 above. Where does the hereditary normality fail? For #1 and #2, $X$ is a compactification of a non-normal space and thus not hereditarily normal. So the dimension for the failure of hereditary normality is 1 for #1 and #2.

For #3 through #7, $X$ is hereditarily normal. For #3 through #5, each $X$ has a closed subset that is not a $G_\delta$ set (hence not perfectly normal). In #3 and #4, the non- $G_\delta$ -set is a single point. In #5, the the non- $G_\delta$ -set is the inner circle. Thus the compact space $X$ in #3 through #5 is not perfectly normal. By Theorem 2, the dimension for the failure of hereditary normality is 2 for #3 through #5.

For #6 and #7, each $X^2$ contains a copy of the Sorgenfrey plane. Thus the dimension for the failure of hereditary normality is also 2 for #6 and #7.

In the small sample of compact non-metrizable spaces just highlighted, the failure of hereditary normality occurs in “dimension” 1 or 2. Naturally, one can ask:

Question

$X$

$X^2$

$X^3$

Such a space $X$ would be an example to show that the condition “ $X^3$ is hereditarily normal” in Theorem 3 is necessary. In other words, the hypothesis in Theorem 3 cannot be weakened if the example just described were to exist.

The above list of compact non-metrizable spaces is a small one. They are fairly standard examples for compact non-metrizable spaces. Could there be some esoteric example out there that fits the description? It turns out that there are such examples. In [1], Gruenhage and Nyikos constructed a compact non-metrizable $X$ such that $X^2$ is hereditarily normal. The construction was done using MA + not CH (Martin’s Axiom coupled with the negation of the continuum hypothesis). In that same paper, they also constructed another another example using CH. With the examples from [1], one immediate question was whether the additional set-theoretic axioms of MA + not CH (or CH) was necessary. Could a compact non-metrizable $X$ such that $X^2$ is hereditarily normal be still constructed without using any axioms beyond ZFC, the generally accepted axioms of set theory? For a relatively short period of time, this was an open question.

In 2001, Larson and Todorcevic [3] showed that it is consistent with ZFC that every compact $X$ with hereditarily normal $X^2$ is metrizable. In other words, there is a model of set theory that is consistent with ZFC in which Theorem 3 can be improved to assuming $X^2$ is hereditarily normal. Thus it is impossible to settle the above question without assuming additional axioms beyond those of ZFC. This means that if a compact non-metrizable $X$ is constructed without using any axiom beyond ZFC (such as those in the small list above), the hereditary normality must fail at dimension 1 or 2. Numerous other examples can be added to the above small list. Looking at these ZFC examples can help us appreciate the results in [1] and [3]. These ZFC examples are excellent training ground for general topology.

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Reference

Gruenhage G., Nyikos P. J., Normality in $X^2$ for Compact $X$ , Trans. Amer. Math. Soc., Vol 340, No 2 (1993), 563-586
Katetov M., Complete normality of Cartesian products, Fund. Math., 35 (1948), 271-274
Larson P., Todorcevic S., KATETOV’S PROBLEM, Trans. Amer. Math. Soc., Vol 354, No 5 (2001), 1783-1791

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$\copyright \ 2015 \text{ by Dan Ma}$

Dan Ma's Topology Blog

Expository Articles In General Topology

Tag Archives: Compact space

Perfect preimages of Lindelof spaces

Making sense of the spaces with small diagonal

Helly Space

Looking for spaces in which every compact subspace is metrizable

The product of locally compact paracompact spaces

An exercise gleaned from the proof of a theorem on pseudocompact space

Products of compact spaces with countable tightness

Normality in the powers of countably compact spaces

The product of uncountably many factors is never hereditarily normal

Looking for non-normal subspaces of the square of a compact X