---
tag: Q6XNC8JAZD
kind: concept
name: Compactification
msc: [54D35]
sources:
  - url: https://en.wikipedia.org/wiki/Compactification_(mathematics)
same_as:
  wikidata: Q546145
  wikipedia: Compactification (mathematics)
---

## Summary

A compactification of a [[topological space|0CQHFPGJ45]] $X$ is a [[compact space|0TVJZYB6SH]] that contains a copy of $X$ as a [[dense|0BSJSX7ME2]] subspace.
It adds "points at infinity" so that sequences or nets that escape $X$ get limits.
Standard examples are the [[one-point compactification|0GWN3CMRK8]] and the [[Stone–Čech compactification|D9MWP5C85W]].
A space has a Hausdorff compactification exactly when it is a [[Tychonoff space|C5N2J2K6Z3]].

## Explanations

### L1 human

The open interval $(0, 1)$ is not compact, because points can run toward the missing ends.
Adding the two end points gives $[0, 1]$, a compactification of $(0, 1)$.
Bending the real line into a circle and adding one point at the top gives another compactification, of $\R$.
Different compactifications of the same space add different sets of ideal points.