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

Symbolic statement

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

Related results

Other results using the same proof technique:

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

Results used in proof

The key result used is the tube lemma, which exploits the compactness of $X$.

Proof

Let $X$ be compact and $Y$ paracompact. We need to prove that $X\times Y$ is paracompact.

Start off with an open cover of $X\times Y$. For each $y\in Y$, this yields an open cover of $X\times \{y\}$ (treated as a copy of $X$). By compactness, we can choose a finite subcover of the cover at each point, and this finite ... Fill this in later