---
tag: 020Q06TYYH
kind: concept
name: Totally bounded set
aliases: [totally-bounded-space, precompact-set]
msc: ["54E35"]
sources:
  - url: https://en.wikipedia.org/wiki/Totally_bounded_space
    locator: lead section
same_as:
  wikidata: Q1362228
  wikipedia: Totally bounded space
---

## Summary

A subset of a metric space is totally bounded when, for each size, it can be covered by finitely many pieces of that size.
Compact sets are exactly the totally bounded sets that are complete.