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 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.
• $\sigma$-compact spaces (link).
• Paracompact locally compact spaces (link).
• Paracompact $\sigma$-locally compact spaces (link).
• Normal countably compact 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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