Cartesian Product of Compact Set and Non-Compact Set is Non-Compact

Question

Theorem: Let $A$ be a compact set and $B$ be a non-compact set. Then $A\times B$ is non-compact.

I know that if $B$ is non-compact, then there exists an open cover $O$ of $B$ that does not have a finite sub-collection that also covers $B$. How can I use this to construct a cover of $A\times B$ that does not have a finite sub-collection?

Thanks!

Answer 1

HINT: If $\mathscr{U}$ is an open cover of $B$ with no finite subcover, consider $\{A\times U:U\in\mathscr{U}\}$.

Answer 2

The (surjective) projection map $A\times B\to B$ is continuous. So if $A\times B$ is compact, then also $B$ is compact.

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More generally, a product space $\prod_{i}X_i$ is compact if and only if every factor $X_i$ is compact.

The “if” part is Tychonov's theorem, the “only if” part is the argument above.