---
tag: D9MWP5C85W
kind: concept
name: Stone–Čech compactification
msc: [54D35]
specializes: [Q6XNC8JAZD]
sources:
  - url: https://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification
same_as:
  wikidata: Q2033899
  wikipedia: Stone–Čech compactification
---

## Summary

The Stone–Čech compactification $\beta X$ of a [[Tychonoff space|C5N2J2K6Z3]] $X$ is the largest Hausdorff [[compactification|Q6XNC8JAZD]] of $X$.
It is characterised by a universal property.
Every continuous map from $X$ to a compact Hausdorff space extends uniquely to $\beta X$.
It links topology with algebra and set theory, for example through the description of $\beta \N$ by [[ultrafilters|J64Y3JC0RE]].

## Explanations

### L1 human

A compactification adds ideal points to make a space compact.
The Stone–Čech compactification adds as many points as possible, in a precise sense.
Every compact Hausdorff compactification of $X$ is a quotient of $\beta X$.
For the natural numbers, the new points of $\beta \N$ correspond to the free ultrafilters on $\N$.