---
tag: 0EQMWQ4WMJ
kind: definition
name: Locally compact space
msc: [54D45]
defines: 0FE2SB8WVJ
uses: [0CNQXVBHCX, 09CJ00H3NW, 00R0X1WGPX]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_compact_space
    locator: Formal definition
---

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is locally compact when each point $x \in X$ has a [[neighbourhood|09CJ00H3NW]] $N$ that is [[compact|00R0X1WGPX]].

## Remarks

Authors use several stronger definitions, for example that each neighbourhood of $x$ contains a compact neighbourhood of $x$. For [[Hausdorff spaces|0MXB4CKHFW]], all the usual definitions agree.
Each compact space is locally compact. The space $\mathbb{R}^n$ is locally compact, because each closed ball is compact.
The rationals $\mathbb{Q}$ and each infinite-dimensional normed space are not locally compact.
A locally compact Hausdorff space has a Hausdorff [[one-point compactification|0MM45BT6HC]].

## Explanations

### L1 human

A locally compact space is compact in small pieces: each point has a compact set around it.
The real line is not compact, but each closed interval in it is.