Stone–Čech compactification

Builds on Topological space Compact space Hausdorff space Continuous map Compactification

Definition

Let $X$ be a topological space. A Stone–Čech compactification of $X$ is a pair $(\beta X, \iota)$ with the following properties.

$\beta X$ is a compact Hausdorff space.
$\iota \colon X \to \beta X$ is a continuous map.
For every compact Hausdorff space $K$ and every continuous map $f \colon X \to K$ , there is a unique continuous map $g \colon \beta X \to K$ with $g \circ \iota = f$ .

Remarks

Such a pair exists for every $X$ and is unique up to a unique homeomorphism compatible with $\iota$ . The map $\iota$ is an embedding with dense image exactly when $X$ is a Tychonoff space. Then $(\beta X, \iota)$ is a compactification of $X$ .

Example: if $X$ is compact Hausdorff, then $\beta X = X$ with $\iota$ the identity. For a continuous $f \colon X \to K$ , the map $g = f$ is the required extension, and it is unique.

Non-example: the one-point compactification $\N \cup \set{\infty}$ of $\N$ is not $\beta \N$ . The map $f \colon \N \to \set{0, 1}$ , $f(n) = n \bmod 2$ , has no continuous extension. An extension would need $g(\infty) = \lim g(2n) = 0$ and $g(\infty) = \lim g(2n + 1) = 1$ .

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`MJP3GW3HD7`
URI	`m2://MJP3GW3HD7`
Kind	definition
Writing	W0 stub
MSC	54D35
Defines	Stone–Čech compactification
Uses	Topological space, Compact space, Hausdorff space, Continuous map, Compactification
Changed	2 November 2026

Stone–Čech compactification

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here