---
tag: ZW4NFH5RNC
kind: concept
name: Sigma-compact space
msc: [54D45]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Σ-compact_space
same_as:
  wikidata: Q290618
  wikipedia: Σ-compact space
---

## Summary

A σ-compact space is a topological space that is a countable union of compact subspaces.
Euclidean space $\R^n$ is the main example, as the union of the closed balls of radius $n$.
Every σ-compact space is [[Lindelöf|SW0SN1NEJE]], and every [[locally compact|0FE2SB8WVJ]] second-countable space is σ-compact.
The notion lets compactness arguments run one piece at a time.

## Explanations

### L1 human

A compact space is small in a topological sense.
A σ-compact space may be big, but it can be built from countably many compact pieces.
The real line is not compact, but it is the union of the intervals $[-n, n]$ for $n = 1, 2, 3, \ldots$.
So many facts proved on compact intervals can be extended to all of $\R$.