# 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 $\mathcal{U}$.