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