---
tag: 0Z4V98VN5E
kind: definition
name: Proper metric space
msc: ["54E45"]
defines: 0RN765S7DH
uses: [05DSV40EB9, 00R0X1WGPX]
sources:
  - url: https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem
    locator: In the theory of metric spaces
---

## Definition

A [[metric space|063W5JMY85]] $(X, d)$ is proper when each [[closed ball|05DSV40EB9]] $\bar{B}(x, r)$, with $x \in X$ and $r \ge 0$, is [[compact|00R0X1WGPX]].

## Remarks

In a proper metric space, a subset is compact if and only if it is closed and bounded. Thus the [[Heine–Borel theorem|07PFYV65MB]] says that $\mathbb{R}^n$ is proper.
Each proper metric space is complete and [[locally compact|0EQMWQ4WMJ]].
An infinite set with the [[discrete metric|0NA4FK7B8S]] is not proper: the closed ball of radius $1$ is the whole set, which is not compact.

## Explanations

### L1 human

A proper metric space is a space where the Heine–Borel test for compactness works: closed and bounded is enough.
Euclidean spaces are proper, but many infinite-dimensional spaces are not.