---
tag: 09ZKE5XQ74
kind: statement
role: theorem
name: Product of compact sets is compact
msc: [54D30]
uses: [0P6PMJB4MA, 0Z57EC6YX3]
sources:
  - url: https://en.wikipedia.org/wiki/Compact_space
    locator: Properties of compact spaces
---

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], and let $K \subseteq X$ and $L \subseteq Y$ be [[compact|00R0X1WGPX]].
Then $K \times L$ is compact in $X \times Y$ with the [[product topology|0Z57EC6YX3]].

## 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|0JTRSJKCFB]].

## 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.