The product of locally compact paracompact spaces

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

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

  1. If Y is a compact space, then X and Y is paracompact.
  2. If Y is a \sigma-compact space, then X and Y is paracompact.
  3. If Y is a locally compact space, then X and Y is paracompact.
  4. If Y is a \sigma-locally compact space, then X and Y is paracompact.

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

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Proof of the Fourth Result

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Productively Paracompact Spaces

Consider this property for a space X.

    (*) The space X satisfies the property that X \times Y is a paracompact space for every paracompact space Y.

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

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

  1. Compact spaces.
  2. \sigma-compact spaces.
  3. Locally compact paracompact spaces.
  4. \sigma-locally compact paracompact spaces.

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

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Normal P-Spaces

Consider this property.

    (**) The space X satisfies the property that X \times Y is a normal space for every metric space Y.

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

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

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

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Looking at Diagrams

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

    Diagram 1

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

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

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

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

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

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

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

Let’s add one more property to Diagram 1.

    Diagram 2

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

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

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Reference

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

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\copyright 2017 – Dan Ma

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