---
tag: 0JTRSJKCFB
kind: statement
role: theorem
name: Tychonoff's theorem
aliases: [tychonoff-theorem]
msc: [54D30]
uses: [04S9HX3GAP, 0QPHW703PV, 09ZKE5XQ74]
sources:
  - url: https://en.wikipedia.org/wiki/Tychonoff%27s_theorem
    locator: Statement
  - url: https://en.wikipedia.org/wiki/Tychonoff%27s_theorem
    locator: Strength
  - url: https://en.wikipedia.org/wiki/Tychonoff%27s_theorem
    locator: lead section
same_as:
  wikidata: Q1137014
  wikipedia: Tychonoff's theorem
---

## Statement

Let $(X_i)_{i \in I}$ be a family of [[compact spaces|0QPHW703PV]], indexed by any set $I$.
Then the product $\prod_{i \in I} X_i$, with the [[product topology|04S9HX3GAP]], is compact.

## Remarks

For finitely many factors, the statement follows from [[product of compact sets is compact|09ZKE5XQ74]].
For arbitrary products, the theorem is equivalent to the axiom of choice.
The statement fails for the [[box topology|0CTM1EHG9P]]: 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.