---
tag: 0RN765S7DH
kind: concept
name: Proper metric space
aliases: [heine-borel-metric-space]
msc: ["54E45"]
sources:
  - url: https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem
    locator: In the theory of metric spaces
---

## Summary

A proper metric space is a metric space in which each closed ball is compact.
In such a space, a set is compact if and only if it is closed and bounded.