Compact times paracompact implies paracompact

This article states and proves a result of the following form: the product of two topological spaces, the first satisfying the property Compact space (?) and the second satisfying the property Paracompact space (?), is a topological space satisfying the property Paracompact space (?).
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Statement

Verbal statement

The product of a compact space with a paracompact space (given the product topology), is paracompact

Statement with symbols

Let $X$ be a compact space and $Y$ a paracompact space. Then $X\times Y$ is paracompact.

Related facts

Other results using the same proof technique:

• Compact times metacompact implies metacompact
• Compact times orthocompact implies orthocompact
• Compact times Lindelof implies Lindelof

Facts used

1. Tube lemma: If $X$ is a compact space and $Y$ is a topological space. Then, given any open subset $U$ of $X\times Y$ containing $X\times \{y\}$ for some $y\in Y$, there exists an open subset $V$ of $Y$ such that $y\in V$ and $X\times V\subseteq U$.

Proof

Given: A compact space $X$, a paracompact space $Y$.

To prove If $U_i$ form an open cover of $X\times Y$, there exists a locally finite open refinement of the $U_i$.

Proof:

1. For any point $y\in Y$, there is a finite collection of $U_i$ that cover $X\times \{y\}$: Since $X$ is compact, the subspace $X\times \{y\}$ of $X\times Y$ is also compact, so the cover by the open subsets $U_i$ has a finite subcover.
2. Let $W_y$ be the union of this finite collection of open subsets $U_i$. By fact (1), there exists an open subset $V_y$ of $Y$ such that $X\times V_y\subseteq W_y$.
3. The $V_y$ form an open cover of $Y$.
4. There exists a locally finite open refinement, say $\mathcal{P}$ of the $V_y$ in $Y$: This follows from the fact that $Y$ is paracompact.
5. We can construct a locally finite open refinement of $U_i$ from these:
   1. For each member $P\in \mathcal{P}$, there exists $V_y$ such that $P\subseteq V_y$. Thus, $X\times P\subseteq X\times V_y\subseteq W_y$. $W_y$, in turn, is a union of a finite collection of $U_i$s. Thus, $X\times P$ is the union of the intersections $(X\times P)\cap U_i$.
   2. Since the $X\times P$ together cover $X\times Y$, the $(X\times P)\cap U_i$ are an open cover of $X\times Y$ that refines the $U_i$s.
   3. Finally, we argue that $(X\times P)\cap U_i$ is a locally finite open cover: Suppose $(x,y)\in X\times Y$. Since $\mathcal{P}$ is a locally finite open cover of $Y$, there exists an open subset $Q$ of $Y$ containing $y$ such that $Q$ intersects only finitely many members of $\mathcal{P}$. Thus, the neighborhood $X\times Q$ intersects only finitely many $X\times P$s, which in turn give rise to finitely many $(X\times P)\cap U_i$s each. Thus, $X\times Q$ intersects only finitely many members of the open cover.