Three conjectures of K Morita

This post discusses the three conjectures that were proposed by K. Morita in 1976. These conjectures concern normality in product spaces. To start the discussion, here’s the conjectures.

Morita’s Conjecture I. Let X be a space. The product X \times Y is normal for every normal space Y if and only if X is a discrete space.

Morita’s Conjecture II. Let X be a space. The product X \times Y is normal for every normal P-space Y if and only if X is a metrizable space.

Morita’s Conjecture III. Let X be a space. The product X \times Y is normal for every normal countably paracompact space Y if and only if X is a metrizable \sigma-locally compact space.

These statements are no longer conjectures. Partial results appeared after the conjectures were proposed in 1976. The complete resolution of the conjectures came in 2001 in a paper by Zoli Balogh [5]. Though it is more appropriate to call these statements theorems, it is still convenient to call them conjectures. Just know that they are now known results rather open problems to be solved. The focus here is not on the evolution of the solutions. Instead, we discuss the relations among the three conjectures and why they are amazing results in the study of normality in product spaces.

As discussed below, in each of these conjectures, one direction is true based on prior known theorems (see Theorem 1, Theorem 2 and Theorem 4 below). The conjectures can be stated as follows.

Morita’s Conjecture I. Let X be a space. If the product X \times Y is normal for every normal space Y then X is a discrete space.

Morita’s Conjecture II. Let X be a space. If the product X \times Y is normal for every normal P-space Y then X is a metrizable space.

Morita’s Conjecture III. Let X be a space. If the product X \times Y is normal for every normal countably paracompact space Y then X is a metrizable \sigma-locally compact space.

P-spaces are defined by K. Morita [11]. He proved that a space X is a normal P-space if and only if the product X \times Y is normal for every metrizable space Y (see theorem 2 below). Normal P-spaces are also discussed here. A space X is \sigma-locally compact space if X is the union of countably many locally compact subspaces each of which is also closed subspace of X.

As we will see below, these conjectures are also called duality conjectures because they are duals of known results.

[2] is a survey of Morita’s conjecture.

Duality Conjectures

Here’s three theorems that are duals to the conjectures.

Theorem 1
Let X be a space. The product space X \times Y is normal for every discrete space Y if and only if X is normal.

Theorem 2
Let X be a space. The product space X \times Y is normal for every metrizable space Y if and only if X is a normal P-space.

Theorem 3
Let X be a space. The product space X \times Y is normal for every metrizable \sigma-locally compact space Y if and only if X is normal countably paracompact.

The key words in red are for emphasis. In each of these three theorems, if we switch the two key words in red, we would obtain the statements for the conjectures. In this sense, the conjectures are called duality conjectures since they are duals of known results.

Theorem 1 is actually not found in the literature. It is an easy theorem. Theorem 2, found in [11], is a characterization of normal P-space (discussed here). Theorem 3 is a well known result based on the following theorem by K. Morita [10].

Theorem 4
Let Y be a metrizable space. Then the product X \times Y is normal for every normal countably paracompact space X if and only if Y is a \sigma-locally compact space.

We now show that Theorem 3 can be established using Theorem 4. Theorem 4 is also Theorem 3.5 in p. 111 of [2]. A proof of Theorem 4 is found in Theorem 1.8 in p. 130 of [8].

Proof of Theorem 3
\Longleftarrow Suppose X is normal and countably paracompact. Let Y be a metrizable \sigma-locally compact space. By Theorem 4, X \times Y is normal.

\Longrightarrow This direction uses Dowker’s theorem. We give a contrapositive proof. Suppose that X is not both normal and countably paracompact. Case 1. X is not normal. Then X \times \{ y \} is not normal where \{ y \} is any one-point discrete space. Case 2. X is normal and not countably paracompact. This means that X is a Dowker space. Then X \times [0,1] is not normal. In either case, X \times Y is not normal for some compact metric space. Thus X \times Y is not normal for some \sigma-locally compact metric space. This completes the proof of Theorem 3. \square

The First and Third Conjectures

The first conjecture of Morita was proved by Atsuji [1] and Rudin [13] in 1978. The proof in [13] is a constructive proof. The key to that solution is to define a \kappa-Dowker space. Suppose X is a non-discrete space. Let \kappa be the least cardinal of a non-discrete subspace of X. Then construct a \kappa-Dowker space Y as in [13]. It follows that X \times Y is not normal. The proof that X \times Y is not normal is discussed here.

Conjecture III was confirmed by Balogh in 1998 [4]. We show here that the first and third conjectures of Morita can be confirmed by assuming the second conjecture.

Conjecture II implies Conjecture I
We give a contrapositive proof of Conjecture I. Suppose that X is not discrete. We wish to find a normal space Y such that X \times Y is not normal. Consider two cases for X. Case 1. X is not metrizable. By Conjecture II, X \times Y is not normal for some normal P-space Y. Case 2. X is metrizable. Since X is infinite and metric, X would contain an infinite compact metric space S. For example, X contains a non-trivial convergent sequence and let S be a convergence sequence plus the limit point. Let Y be a Dowker space. Then the product S \times Y is not normal. It follows that X \times Y is not normal. Thus there exists a normal space Y such that X \times Y is not normal in either case. \square

Conjecture II implies Conjecture III
Suppose that the product X \times Y is normal for every normal and countably paracompact space Y. Since any normal P-space is a normal countably paracompact space, X \times Y is normal for every normal and P-space Y. By Conjecture II, X is metrizable. By Theorem 4, X is \sigma-locally compact. \square

The Second Conjecture

The above discussion shows that a complete solution to the three conjectures hinges on the resolution of the second conjecture. A partial resolution came in 1986 [6]. In that paper, it was shown that under V = L, conjecture II is true.

The complete solution of the second conjecture is given in a paper of Balogh [5] in 2001. The path to Balogh’s proof is through a conjecture of M. E. Rudin identified as Conjecture 9.

Rudin’s Conjecture 9. There exists a normal P-space X such that some uncountable increasing open cover of X cannot be shrunk.

Conjecture 9 was part of a set of 14 conjectures stated in [14]. It is also discussed in [7]. In [6], conjecture 9 was shown to be equivalent to Morita’s second conjecture. In [5], Balogh used his technique for constructing a Dowker space of cardinality continuum to obtain a space as described in conjecture 9.

The resolution of conjecture II is considered to be one of Balogh greatest hits [3].

Abundance of Non-Normal Products

One immediate observation from Morita’s conjecture I is that existence of non-normal products is wide spread. Conjecture I indicates that every normal non-discrete space X is paired with some normal space Y such that their product is not normal. So every normal non-discrete space forms a non-normal product with some normal space. Given any normal non-discrete space (no matter how nice it is or how exotic it is), it can always be paired with another normal space (sometimes paired with itself) for a non-normal product.

Suppose we narrow the focus to spaces that are normal and non-metrizable. Then any such space X is paired with some normal P-space Y to form a non-normal product space (Morita’s conjecture II). By narrowing the focus on X to the non-metrizable spaces, we obtain more clarity on the paired space to form non-normal product, namely a normal P-space. As an example, let X be the Michael line (normal and non-metrizable). It is well known that X in this case is paired with \mathbb{P}, the space of irrational numbers with the usual Euclidean topology, to form a non-normal product (discussed here).

Another example is X being the Sorgenfrey line. It is well known that X in this case is paired with itself to form a non-normal product (discussed here). Morita’s conjectures are powerful indication that these two non-normal products are not isolated phenomena.

Another interesting observation about conjecture II is that normal P-spaces are not productive with respect to normality. More specifically, for any non-metrizable normal P-space X, conjecture II tells us that there exists another normal P-space Y such that X \times Y is not normal.

Now we narrow the focus to spaces that are metrizable but not \sigma-locally compact. For any such space X, conjecture III tells us that X is paired with a normal countably paracompact space Y to form a non-normal product. Using the Michael line example, this time let X=\mathbb{P}, the space of irrational numbers, which is a metric space that is not \sigma-locally compact. The paired normal and countably paracompact space Y is the Michael line.

Each conjecture is about existence of a normal Y that is paired with a given X to form a non-normal product. For Conjecture I, the given X is from a wide class (normal non-discrete). As a result, there is not much specific information on the paired Y, other than that it is normal. For Conjectures II and III, the given space X is from narrower classes. As a result, there is more information on the paired Y.

The concept of Dowker spaces runs through the three conjectures, especially the first conjecture. Dowker spaces and \kappa-Dowker spaces provide reliable pairing for non-normal products. In fact this is one way to prove conjecture I [13], also see here. For any normal space X with a countable non-discrete subspace, the product of X and any Dowker space is not normal (discussed here). For any normal space X such that the least cardinality of a non-discrete subspace is an uncountable cardinal \kappa, the product X \times Y is not normal where Y is a \kappa-Dowker space as constructed in [13], also discussed here.

In finding a normal pair Y for a normal space X, if we do not care about Y having a high degree of normal productiveness (e.g. normal P or normal countably paracompact), we can always let Y be a Dowker space or \kappa-Dowker space. In fact, if the starting space X is a metric space, the normal pair for a non-normal product (by definition) has to be a Dowker space. For example, if X=[0,1], then the normal space Y such that X \times Y is by definition a Dowker space. The search for a Dowker space spanned a period of 20 years. For the real line \mathbb{R}, the normal pair for a non-normal product is also a Dowker space. For “nice” spaces such as metric spaces, finding a normal space to form non-normal product is no trivial problem.

Reference

  1. Atsuji M.,On normality of the product of two spaces, General Topology and Its Relation to Modern Analysis and Algebra (Proc. Fourth Prague Topology sympos., 1976), Part B, 25–27, 1977.
  2. Atsuji M.,Normality of product spaces I, in: K. Morita, J. Nagata (Eds.), Topics in General
    Topology, North-Holland, Amsterdam, 81–116, 1989.
  3. Burke D., Gruenhage G.,Zoli, Top. Proc., Vol. 27, No 1, i-xxii, 2003.
  4. Balogh Z.,Normality of product spaces and K. Morita’s third conjecture, Topology Appl., Vol. 84, 185-198, 1998.
  5. Balogh Z.,Normality of product spaces and Morita’s conjectures, Topology Appl., Vol. 115, 333-341, 2001.
  6. Chiba K., Przymusinski T., Rudin M. E.Nonshrinking open covers and K. Morita’s duality conjectures, Topology Appl., Vol. 22, 19-32, 1986.
  7. Gruenhage G.,Mary Ellen’s Conjectures,, Special Issue honoring the memory of Mary Ellen Rudin, Topology Appl., Vol. 195, 15-25, 2015.
  8. Hoshina T.,Normality of product spaces II, in: K. Morita, J. Nagata (Eds.), Topics in General Topology, North-Holland, Amsterdam, 121–158, 1989.
  9. Morita K., On the Product of a Normal Space with a Metric Space, Proc. Japan Acad., Vol. 39, 148-150, 1963. (article information; paper)
  10. Morita K., Products of Normal Spaces with Metric Spaces II, Sci. Rep. Tokyo Kyoiku Dagaiku Sec A, 8, 87-92, 1963.
  11. Morita K., Products of Normal Spaces with Metric Spaces, Math. Ann., Vol. 154, 365-382, 1964.
  12. Morita K., Nagata J., Topics in General Topology, Elsevier Science Publishers, B. V., The Netherlands, 1989.
  13. Rudin M. E., \kappa-Dowker Spaces, Czechoslovak Mathematical Journal, 28, No.2, 324-326, 1978.
  14. Rudin M. E., Some conjectures, in: Open Problems in Topology, J. van Mill and G.M. Reed,
    eds., North Holland, 184–193, 1990.
  15. Telgárski R., A characterization of P-spaces, Proc. Japan Acad., Vol. 51, 802–807, 1975.

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Dan Ma math

Daniel Ma mathematics

\copyright 2018 – Dan Ma

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Morita’s normal P-space

In this post we discuss K. Morita’s notion of P-space, which is a useful and interesting concept in the study of normality of product spaces.

The Definition

In [1] and [2], Morita defined the notion of P-spaces. First some notations. Let \kappa be a cardinal number such that \kappa \ge 1. Conveniently, \kappa is identified by the set of all ordinals preceding \kappa. Let \Gamma be the set of all finite sequences (\alpha_1,\alpha_2,\cdots,\alpha_n) where n=1,2,\cdots and all \alpha_i < \kappa. Let X be a space. The collection \left\{A_\sigma \subset X: \sigma \in \Gamma \right\} is said to be decreasing if this condition holds: for any \sigma \in \Gamma and \delta \in \Gamma with

    \sigma =(\alpha_1,\alpha_2,\cdots,\alpha_n)

    \delta =(\beta_1,\beta_2,\cdots,\beta_n, \cdots, \beta_m)

such that n<m and such that \alpha_i=\beta_i for all i \le n, we have A_{\delta} \subset A_{\sigma}. On the other hand, the collection \left\{A_\sigma \subset X: \sigma \in \Gamma \right\} is said to be increasing if for any \sigma \in \Gamma and \delta \in \Gamma as described above, we have A_{\sigma} \subset A_{\delta}.

The space X is a P-space if for any cardinal \kappa \ge 1 and for any decreasing collection \left\{F_\sigma \subset X: \sigma \in \Gamma \right\} of closed subsets of X, there exists open set U_\sigma for each \sigma \in \Gamma with F_\sigma \subset U_\sigma such that for any countably infinite sequence (\alpha_1,\alpha_2,\cdots,\alpha_n,\cdots) where each finite subsequence \sigma_n=(\alpha_1,\alpha_2,\cdots,\alpha_n) is an element of \Gamma, if \bigcap_{n=1}^\infty F_{\sigma_n}=\varnothing, then \bigcap_{n=1}^\infty U_{\sigma_n}=\varnothing.

By switching closed sets and open sets and by switching decreasing collection and increasing collection, the following is an alternative but equivalent definition of P-spaces.

The space X is a P-space if for any cardinal \kappa \ge 1 and for any increasing collection \left\{U_\sigma \subset X: \sigma \in \Gamma \right\} of open subsets of X, there exists closed set F_\sigma for each \sigma \in \Gamma with F_\sigma \subset U_\sigma such that for any countably infinite sequence (\alpha_1,\alpha_2,\cdots,\alpha_n,\cdots) where each finite subsequence \sigma_n=(\alpha_1,\alpha_2,\cdots,\alpha_n) is an element of \Gamma, if \bigcup_{n=1}^\infty U_{\sigma_n}=X, then \bigcup_{n=1}^\infty F_{\sigma_n}=X.

Note that the definition is per cardinal number \kappa \ge 1. To bring out more precision, we say a space X is a P(\kappa)-space of it satisfies the definition for P-space for the cardinal \kappa. Of course if a space is a P(\kappa)-space for all \kappa \ge 1, then it is a P-space.

There is also a game characterization of P-spaces [4].

A Specific Case

It is instructive to examine a specific case of the definition. Let \kappa=1=\{ 0 \}. In other words, let’s look what what a P(1)-space looks like. The elements of the index set \Gamma are simply finite sequences of 0’s. The relevant information about an element of \Gamma is its length (i.e. a positive integer). Thus the closed sets F_\sigma in the definition are essentially indexed by integers. For the case of \kappa=1, the definition can be stated as follows:

For any decreasing sequence F_1 \supset F_2 \supset F_3 \cdots of closed subsets of X, there exist U_1,U_2,U_3,\cdots, open subsets of X, such that F_n \subset U_n for all n and such that if \bigcap_{n=1}^\infty F_n=\varnothing then \bigcap_{n=1}^\infty U_n=\varnothing.

The above condition implies the following condition.

For any decreasing sequence F_1 \supset F_2 \supset F_3 \cdots of closed subsets of X such that \bigcap_{n=1}^\infty F_n=\varnothing, there exist U_1,U_2,U_3,\cdots, open subsets of X, such that F_n \subset U_n for all n and such that \bigcap_{n=1}^\infty U_n=\varnothing.

The last condition is one of the conditions in Dowker’s Theorem (condition 6 in Theorem 1 in this post and condition 7 in Theorem 1 in this post). Recall that Dowker’s theorem states that a normal space X is countably paracompact if and only if the last condition holds if and only of the product X \times Y is normal for every infinite compact metric space Y. Thus if a normal space X is a P(1)-space, it is countably paracompact. More importantly P(1) space is about normality in product spaces where one factor is a class of metric spaces, namely the compact metric spaces.

Based on the above discussion, any normal space X that is a P-space is a normal countably paracompact space.

The definition for P(1)-space is identical to one combinatorial condition in Dowker’s theorem which says that any decreasing sequence of closed sets with empty intersection has an open expansion that also has empty intersection.

For P(\kappa)-space where \kappa>1, the decreasing family of closed sets are no longer indexed by the integers. Instead the decreasing closed sets are indexed by finite sequences of elements of \kappa. The index set \Gamma would be more like a tree structure. However the look and feel of P-space is like the combinatorial condition in Dowker’s theorem. The decreasing closed sets are expanded by open sets. For any “path in the tree” (an infinite sequence of elements of \kappa), if the closed sets along the path has empty intersection, then the corresponding open sets would have empty intersection.

Not surprisingly, the notion of P-spaces is about normality in product spaces where one factor is a metric space. In fact, this is precisely the characterization of P-spaces (see Theorem 1 and Theorem 2 below).

A Characterization of P-Space

Morita gave the following characterization of P-spaces among normal spaces. The following theorems are found in [2].

Theorem 1
Let X be a space. The space X is a normal P-space if and only if the product space X \times Y is normal for every metrizable space Y.

Thus the combinatorial definition involving decreasing families of closed sets being expanded by open sets is equivalent to a statement that is much easier to understand. A space that is normal and a P-space is precisely a normal space that is productively normal with every metric space. The following theorem is Theorem 1 broken out for each cardinal \kappa.

Theorem 2
Let X be a space and let \kappa \ge \omega. Then X is a normal P(\kappa)-space if and only if the product space X \times Y is normal for every metric space Y of weight \kappa.

Theorem 2 only covers the infinite cardinals \kappa starting with the countably infinite cardinal. Where are the P(n)-spaces placed where n are the positive integers? The following theorem gives the answer.

Theorem 3
Let X be a space. Then X is a normal P(2)-space if and only if the product space X \times Y is normal for every separable metric space Y.

According to Theorem 2, X is a normal P(\omega)-space if and only if the product space X \times Y is normal for every separable metric space Y. Thus suggests that any P(2)-space is a P(\omega)-space. It seems to say that P(2) is identical to P(\kappa) where \kappa is the countably infinite cardinal. The following theorem captures the idea.

Theorem 4
Let \kappa be the positive integers 2,3,4,\cdots or \kappa=\omega, the countably infinite cardinal. Let X be a space. Then X is a P(2)-space if and only if X is a P(\kappa)-space.

To give a context for Theorem 4, note that if X is a P(\kappa)-space, then X is a P(\tau)-space for any cardinal \tau less than \kappa. Thus if X is a P(3)-space, then it is a P(2)-space and also a P(1)-space. In the definition of P(\kappa)-space, the index set \Gamma is the set of all finite sequences of elements of \kappa. If the definition for P(\kappa)-space holds, it would also hold for the index set consisting of finite sequences of elements of \tau where \tau<\kappa. Thus if the definition for P(\omega)-space holds, it would hold for P(n)-space for all integers n.

Theorem 4 says that when the definition of P(2)-space holds, the definition would hold for all larger cardinals up to \omega.

In light of Theorem 1 and Dowker's theorem, we have the following corollary. If the product of a space X with every metric space is normal, then the product of X with every compact metric space is normal.

Corollary 5
Let X be a space. If X is a normal P-space, then X is a normal and countably paracompact space.

Examples of Normal P-Space

Here’s several classes of spaces that are normal P-spaces.

  • Metric spaces.
  • Compact spaces (link).
  • \sigma-compact spaces (link).
  • Paracompact locally compact spaces (link).
  • Paracompact \sigma-locally compact spaces (link).
  • Normal countably compact spaces (link).
  • Perfectly normal spaces (link).
  • \Sigma-product of real lines.

Clearly any metric space is a normal P-space since the product of any two metric spaces is a metric space. Any compact space is a normal P-space since the product of a compact space and a paracompact space is paracompact, hence normal. For each of the classes of spaces listed above, the product with any metric space is normal. See the corresponding links for proofs of the key theorems.

The \Sigma-product of real lines \Sigma_{\alpha<\tau} \mathbb{R} is a normal P-space. For any metric space Y, the product (\Sigma_{\alpha<\tau} \mathbb{R}) \times Y is a \Sigma-product of metric spaces. By a well known result, the \Sigma-product of metric spaces is normal.

Examples of Non-Normal P-Spaces

Paracompact \sigma-locally compact spaces are normal P-spaces since the product of such a space with any paracompact space is paracompact. However, the product of paracompact spaces in general is not normal. The product of Michael line (a hereditarily paracompact space) and the space of irrational numbers (a metric space) is not normal (discussed here). Thus the Michael line is not a normal P-space. More specifically the Michael line fails to be a normal P(2)-space. However, it is a normal P(1)-space (i.e. normal and countably paracompact space).

The Michael line is obtained from the usual real line topology by making the irrational points isolated. Instead of using the irrational numbers, we can obtain a similar space by making points in a Bernstein set isolated. The resulting space X is a Michael line-like space. The product of X with the starting Bernstein set (a subset of the real line with the usual topology) is not normal. Thus this is another example of a normal space that is not a P(2)-space. See here for the details of how this space is constructed.

To look for more examples, look for non-normal product X \times Y where one factor is normal and the other is a metric space.

More Examples

Based on the characterization theorem of Morita, normal P-spaces are very productively normal. Normal P-spaces are well behaved when taking product with metrizable spaces. However, they are not well behaved when taking product with non-metrizable spaces. Let’s look at several examples.

Consider the Sorgenfrey line. It is perfectly normal. Thus the product of the Sorgenfrey line with any metric space is also perfectly normal, hence normal. It is well known that the square of the Sorgenfrey line is not normal.

The space \omega_1 of all countable ordinals is a normal and countably compact space, hence a normal P-space. However, the product of \omega_1 and some compact spaces are not normal. For example, \omega_1 \times (\omega_1 +1) is not normal. Another example: \omega_1 \times I^I is not normal where I=[0,1]. The idea here is that the product of \omega_1 and any compact space with uncountable tightness is not normal (see here).

Compact spaces are normal P-spaces. As discussed in the preceding paragraph, the product of any compact space with uncountable tightness and the space \omega_1 is not normal.

Even as nice a space as the unit interval [0,1], it is not always productive. The product of [0,1] with a Dowker space is not normal (see here).

In general, normality is not preserved in the product space operation. the best we can ask for is that normal spaces be productively normal with respect to a narrow class of spaces. For normal P-spaces, that narrow class of spaces is the class of metric spaces. However, normal product is not a guarantee outside of the productive class in question.

Reference

  1. Morita K., On the Product of a Normal Space with a Metric Space, Proc. Japan Acad., Vol. 39, 148-150, 1963. (article information; paper)
  2. Morita K., Products of Normal Spaces with Metric Spaces, Math. Ann., Vol. 154, 365-382, 1964.
  3. Morita K., Nagata J., Topics in General Topology, Elsevier Science Publishers, B. V., The Netherlands, 1989.
  4. Telgárski R., A characterization of P-spaces, Proc. Japan Acad., Vol. 51, 802–807, 1975.

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Dan Ma math

Daniel Ma mathematics

\copyright 2018 – Dan Ma

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

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

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

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

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

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

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

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

____________________________________________________________________

The Proof

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

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

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

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

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

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

such that

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

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

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

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

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

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

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

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

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Remarks

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

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

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

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

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P-Space in the Sense of Morita

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

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

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

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

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

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Reference

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

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\copyright \ 2017 \text{ by Dan Ma}

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

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

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

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

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

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The Notion of Shrinking

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

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

Theorem 1
The following conditions are equivalent.

  1. The space X is normal.
  2. Every point-finite open cover of X is shrinkable.
  3. Every locally finite open cover of X is shrinkable.
  4. Every finite open cover of X is shrinkable.
  5. Every two-element open cover of X is shrinkable.

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

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

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Key Idea

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

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

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

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

Then \mathcal{U} is shrinkable.

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

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

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

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

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

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

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

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

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

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

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

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

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

Then it follows that

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

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

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

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Main Theorem

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

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

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

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

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

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

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

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

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

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

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

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Remarks

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

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

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Reference

  1. 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.
  2. Stone A. H., Paracompactness and Product Spaces, Bull. Amer. Math. Soc., Vol. 54, 977-982, 1948. (paper)
  3. Yang L., The Normality in Products with a Countably Compact Factor, Canad. Math. Bull., Vol. 41 (2), 245-251, 1998. (abstract, paper)

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\copyright \ 2017 \text{ by Dan Ma}

kappa-Dowker space and the first conjecture of Morita

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

All spaces under consideration are Hausdorff.

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Morita’s First Conjecture

In 1976, K. Morita posed the following conjecture.

    Morita’s First Conjecture
    If Y is a normal space such that X \times Y is a normal space for every normal space X, then Y is a discrete space.

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

    Morita’s First Conjecture
    If Y is a non-discrete normal space, then there exists some normal space X such that X \times Y is not a normal space.

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

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

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Dowker’s Theorem

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

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

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

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

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

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

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

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Shrinking Properties

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

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

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

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

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

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An Attempt for a Generalized Dowker’s Theorem

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

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

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

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

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

  1. The space X is a \kappa-paracompact space.
  2. The space X is a \kappa-shrinking space.
    • For each open cover \left\{U_\alpha: \alpha<\kappa \right\} of X, there exists an open cover \left\{V_\alpha: \alpha<\kappa \right\} such that \overline{V_\alpha} \subset U_\alpha for each \alpha<\kappa.
  3. The space X has property \mathcal{D}(\kappa).
    • For each increasing open cover \left\{U_\alpha: \alpha<\kappa \right\} of X, there exists an open cover \left\{V_\alpha: \alpha<\kappa \right\} such that \overline{V_\alpha} \subset U_\alpha for each \alpha<\kappa.
  4. For each decreasing family \left\{F_\alpha: \alpha<\kappa \right\} of closed subsets of X such that \bigcap_{\alpha<\kappa} F_\alpha=\varnothing, there exists a family \left\{G_\alpha: \alpha<\kappa \right\} of open subsets of X such that \bigcap_{\alpha<\kappa} G_\alpha=\varnothing and F_\alpha \subset G_\alpha for each \alpha<\kappa.
  5. The space X has property \mathcal{B}(\kappa).
    • For each increasing open cover \left\{U_\alpha: \alpha<\kappa \right\} of X, there exists an increasing open cover \left\{V_\alpha: \alpha<\kappa \right\} such that \overline{V_\alpha} \subset U_\alpha for each \alpha<\kappa.
  6. The product space X \times Y_\kappa is a normal space.
  7. The product space X \times Y is a normal space for some space Y with mt(Y)=\kappa.

The following diagram shows how these conditions are related.

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

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

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

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

    3 \Longleftrightarrow 4 (Theorem 4)

    5 \Longleftrightarrow 6 (Theorem 7)

    2 \not \Longrightarrow 5 (Example 1)

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

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Proof of Theorem 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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\kappa-Dowker Space

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

    There exists a decreasing family \left\{F_\alpha: \alpha<\kappa \right\} of closed subsets of X with \bigcap_{\alpha<\kappa} F_\alpha=\varnothing such that for every family \left\{G_\alpha: \alpha<\kappa \right\} of open subsets of X with F_\alpha \subset G_\alpha for each \alpha, \bigcap_{\alpha<\kappa} G_\alpha \ne \varnothing.

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

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

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

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Remarks

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

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

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

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

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

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

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Loose Ends

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

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

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

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

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

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Reference

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

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\copyright \ 2017 \text{ by Dan Ma}

Spaces with shrinking properties

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

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

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Shrinking Spaces

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

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

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Examples of Shrinking

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

Theorem 1
The following conditions are equivalent.

  1. The space X is normal.
  2. Every point-finite open cover of X is shrinkable.
  3. Every locally finite open cover of X is shrinkable.
  4. Every finite open cover of X is shrinkable.
  5. Every two-element open cover of X is shrinkable.

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

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

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

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

Corollary 2

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

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

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

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

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

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

    Diagram 1

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

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Three Shrinking Related Properties

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

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

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

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

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

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

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

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

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

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

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

    Diagram 2

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

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

    Diagram 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Property \mathcal{B}(\kappa)

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

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

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

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

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

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

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

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

  • The set [d_\alpha,p] is an open set in Y_\kappa containing the point p and is defined as follows.
    • [d_\alpha,p]=\left\{d_\beta: \alpha \le \beta<\kappa \right\} \cup \left\{p \right\}.
  • For any two disjoint closed subsets H and K of the product space X \times Y_\kappa, define the following sets.
    • For each \alpha<\kappa, let H_\alpha=H \cap (X \times \left\{d_\alpha \right\}) and K_\alpha=K \cap (X \times \left\{d_\alpha \right\}).
    • Let H_p=H \cap (X \times \left\{p \right\}) and K_p=K \cap (X \times \left\{p \right\}).
    • For each \alpha<\kappa, choose open O_\alpha \subset X such that G_\alpha=O_\alpha \times \left\{d_\alpha \right\}, H_\alpha \subset G_\alpha and \overline{G_\alpha} \cap K_\alpha=\varnothing (due to normality of X).
    • Choose open O_p \subset X such that G_p=O_p \times \left\{p \right\}, H_p \subset G_p and \overline{G_p} \cap K_p=\varnothing (due to normality of X).

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

    Case 1. H \subset X \times D_\kappa and K \subset X \times D_\kappa.
    Case 2a. H=X \times \left\{p\right\}
    Case 2b. Exactly one of H and K intersect the set X \times \left\{p\right\}.
    Case 3. Both H and K intersect the set X \times \left\{p\right\}.

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

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

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

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

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

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

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

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

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

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

Now project H_p and K_p onto the space X.

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

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

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

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

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

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

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

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

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

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

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

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

such that

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

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

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

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

    \overline{W_{\alpha,1}} \subset \overline{V_{\alpha,1}} \subset T_{\alpha,1} \subset U_{\alpha,1} (closure with respect to X)

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

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

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

    H_\alpha \subset L_\alpha

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

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

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

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

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

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

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

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

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

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

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

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Reference

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

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\copyright \ 2017 \text{ by Dan Ma}

Product Space – Exercise Set 1

This post presents several exercises concerning product spaces. All the concepts involved in the exercises have been discussed in the blog.

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Exercise 1

Exercise 1a
Prove or disprove:
If X and Y are both hereditarily separable, then X \times Y is hereditarily separable.

Exercise 1b
Show that if each X_\alpha is separable, then the product space \prod_{\alpha < \omega} \ X_\alpha is separable.

Exercise 1c
Prove or disprove:
If each X_\alpha is separable, then the product space \prod_{\alpha < \omega_1} \ X_\alpha is not separable.

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Exercise 2

Exercise 2a
Show that if the space X is normal, then every closed subspace of X is a normal space.

Exercise 2b
Prove or disprove:
If the space X is normal, then every dense open subspace of X is a normal space.

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Exercise 3

Consider the product space \prod_{\alpha \in W} \ X_\alpha.

Exercise 3a
Suppose that X_\alpha is compact for all but one \alpha \in W such that the non-compact factor is a Lindelof space. Show that the product space \prod_{\alpha \in W} \ X_\alpha is a normal space.

Exercise 3b
Prove or disprove:
Suppose that X_\alpha is compact for all but one \alpha \in W such that the non-compact factor is a normal space. Then the product space \prod_{\alpha \in W} \ X_\alpha is a normal space.

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Exercise 4

Exercise 4a
Let X be a compact space.
Show that if X^n is hereditarily Lindelof for all positive integer n, then X is metrizable.

Exercise 4b
Prove or disprove:
If X^n is hereditarily Lindelof for all positive integer n, then X is metrizable.

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Exercise 5

Let Y the product of uncountably many copies of the real line \mathbb{R}. If a specific example is desired, try Y=\mathbb{R}^{\omega_1} (\omega_1 many copies of \mathbb{R}) or Y=\mathbb{R}^{\mathbb{R}} (continuum many copies of \mathbb{R}). It is also OK to use a larger number of copies of the real line.

Note that the space Y is not normal (see here).

Exercise 5a
Since the product space Y is not normal, it is not Lindelof. As an exercise, find an open cover of Y that proves that Y is not Lindelof, i.e. an open cover \mathcal{U} of Y such that no countable subcollection of \mathcal{U} can cover Y.

Exercise 5b
Show that for every open cover \mathcal{U} of the space Y, there is a countable \mathcal{V} \subset \mathcal{U} of Y such that \overline{\mathcal{V}}=Y, i.e. \cup \mathcal{V} is dense in Y. Note that with this property, the space Y is said to be weakly Lindelof.

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Exercise 6

This exercise is about the product Y=\mathbb{R}^{\mathbb{R}} (continuum many copies of \mathbb{R}). Show the following.

  1. Show that Y is separable by exhibiting a countable dense set.
  2. Show that Y is not hereditarily separable by exhibiting a non-separable subspace.
  3. Show that the space Y has a closed and discrete subspace of cardinality continuum.
  4. Show that Y is not first countable.
  5. Show that Y is not a Frechet space.
  6. Show that Y is not a countably tight space.

See here for the definition of Frechet space.

See here for the definition of countably tight space.

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Exercise 7

Consider the product space Y=\mathbb{\omega}^{\omega_1}. It is not normal (see here).

Exercise 7a
Construct a dense normal subspace of Y.

Exercise 7b
Construct a dense Lindelof subspace of Y.

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\copyright \ 2016 \text{ by Dan Ma}