Product of compact sets is compact

Statement

Let $X$ and $Y$ be topological spaces, and let $K \subseteq X$ and $L \subseteq Y$ be compact. Then $K \times L$ is compact in $X \times Y$ with the product topology.

Remarks

The proof uses the tube lemma. For each $x \in K$, finitely many basic open sets $U \times V$ cover $\{x\} \times L$, and their intersection over the first factor gives an open set about $x$. By induction, a product of finitely many compact sets is compact. For example, $[0, 1]^n$ is compact. For arbitrary products of compact spaces, the statement is Tychonoff's theorem.

Explanations

L1 human

A rectangle made of two compact sides is compact. For example, the square $[0, 1] \times [0, 1]$ is compact, because $[0, 1]$ is compact.