Tychonoff's theorem

Statement

Let $(X_i)_{i \in I}$ be a family of compact spaces, indexed by any set $I$. Then the product $\prod_{i \in I} X_i$, with the product topology, is compact.

Remarks

For finitely many factors, the statement follows from product of compact sets is compact. For arbitrary products, the theorem is equivalent to the axiom of choice. The statement fails for the box topology: the box topology on $\{0, 1\}^{\mathbb{N}}$ is discrete and infinite. For example, $[0, 1]^I$ is compact for each set $I$.

History

Andrey Tikhonov proved the theorem in 1930 for powers of the closed unit interval. He stated the general theorem in 1935, with the remark that the same proof works.

Explanations

L1 human

Any product of compact spaces is compact, even with infinitely many factors. It works because a basic open set of the product restricts only finitely many coordinates.