Compactification

Builds on Topological space Compact space Topological embedding Dense set

Definition

Let $X$ be a topological space. A compactification of $X$ is a pair $(K, e)$ with the following properties.

$K$ is a compact space.
$e \colon X \to K$ is a topological embedding.
The image $e(X)$ is dense in $K$ .

Remarks

Example: $K = [0, 1]$ with the inclusion $e \colon (0, 1) \to [0, 1]$ is a compactification of $(0, 1)$ . The interval $[0, 1]$ is compact, the inclusion is an embedding for the subspace topology, and the closure of $(0, 1)$ is $[0, 1]$ .

Non-example: $K = [0, 2]$ with the inclusion of $(0, 1)$ fails clause 3. The closure of $(0, 1)$ in $[0, 2]$ is $[0, 1]$ , not $[0, 2]$ .

Many authors also require $K$ to be Hausdorff. Two compactifications $(K, e)$ and $(K', e')$ are equivalent if there is a homeomorphism $h \colon K \to K'$ with $h \circ e = e'$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Compact space concept
Dense set concept
Topological embedding concept
Topological space concept
Compactification definition

Tag	`355XPVR564`
URI	`m2://355XPVR564`
Kind	definition
Writing	W0 stub
MSC	54D35
Defines	Compactification
Uses	Topological space, Compact space, Topological embedding, Dense set
Changed	2 November 2026

Compactification

Definition

Remarks

Reading path

Further reading

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