---
tag: 355XPVR564
kind: definition
name: Compactification
msc: [54D35]
defines: Q6XNC8JAZD
uses: [0CQHFPGJ45, 0TVJZYB6SH, 00ZG9Y9TR4, 0BSJSX7ME2]
sources:
  - url: https://en.wikipedia.org/wiki/Compactification_(mathematics)
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A *compactification* of $X$ is a pair $(K, e)$ with the following properties.

1. $K$ is a [[compact space|0TVJZYB6SH]].
2. $e \colon X \to K$ is a [[topological embedding|00ZG9Y9TR4]].
3. The image $e(X)$ is [[dense|0BSJSX7ME2]] 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 $h \circ e = e'$.