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 "{{{left}}}" is not a number. and the second satisfying the property "{{{right}}}" is not a number., is a topological space satisfying the property "{{{final}}}" is not a number..
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This fact is related to: compactness

Statement

Verbal statement

The direct product of a compact space with a paracompact space, 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

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