Compact x Paracompact Is Paracompact

The theorem indicated by the title is a basic result and is an interesting theorem. It is known that “compact x normal” needs not be normal (see an example here). But for a compact factor and a paracompact factor, the product is not only normal, it is paracompact. I would like to write down a proof. This theorem will be used in subsequent posts. If the factor Y is Lindelof instead, it can be shown that X \times Y is Lindelof.

Let X be a compact space and let Y be a paracompact space. Let \mathcal{U} be an open cover of X \times Y. For each y \in Y, let \mathcal{G}_y \subset \mathcal{U} be a finite cover of X \times \lbrace{y}\rbrace.

Claim. Fix y \in Y. There is an open set W_y \subset Y such that X \times W_y \subset \cup \mathcal{G}_y. For each x \in X, let A_x \subset X be open and let B_x \subset Y be open such that x \in A_x, y \in B_x, and A_x \times B_x \subset \cup\mathcal{G}_y. Since X is compact, choose finitely many open sets A_x that cover X, say A_{x(0)},A_{x(1)},...,A_{x(n)}. Let W_y=B_{x(0)} \cap B_{x(1)} \cap...\cap B_{x(n)}. It is clear that X \times W_y \subset \cup \mathcal{G}_y.

The collection \mathcal{W}=\lbrace{W_y:y \in Y}\rbrace is an open cover of Y. Then it has a locally finite open refinement \mathcal{E}. For each E \in\mathcal{E}, choose y \in Y such that E \subset W_y. Consider (X \times E) \cap G where G \in\mathcal{G}_y. Let \mathcal{F} be the collection of all such open sets (X \times E) \cap G.

There are three things to check here. One is that \mathcal{F} is an open cover of X \times Y. Fix (x,z) \in X \times Y. Then z \in E for some E \in\mathcal{E}. Note that X \times E \subset X \times W_y \subset \cup \mathcal{G}_y for some y \in Y. This means that the point (x,z) \in (X \times E)\cap G for some G in \mathcal{G}_y.

The second to check is that \mathcal{F} is a refinement of the original open cover \mathcal{U}. This is clear since every set in \mathcal{F} is chosen to be a subset of some set in \mathcal{G}_y \subset \mathcal{U}.

The third is that \mathcal{F} is a locally finite collection. To see this, let (x,z) \in X \times Y. There is an open V \subset Y such that z \in V and V can only meet finitely many E \in\mathcal{E}. For each such E, X \times E is associated with finitely many sets in \mathcal{F}. This means X \times V is an open set containing (x,z) that can meet only finitely many sets in \mathcal{F}. This completes the proof that X \times Y is paracompact.

If the factor Y is Lindelof, then we can modify the proof to show that X \times Y is Lindelof. In the step above where \mathcal{W}=\lbrace{W_y:y \in Y}\rbrace is obtained, we get a countable subcover of \mathcal{W}. Each member of this countable subcover is associated with a finite \mathcal{G}_y \subset \mathcal{U}. Thus we can obtain a countable subcover of \mathcal{U}.

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