# Definition:Sigma-Compact Space

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

$T$ is $\sigma$-compact if and only if $S$ is the union of the underlying sets of countably many compact subspaces of $T$.

This can be expressed more efficiently as:

$T$ is $\sigma$-compact if and only if it is the union of countably many compact subspaces.

## Also known as

A $\sigma$-compact space is also known as a space that is countable at infinity.

## Also see

• Results about $\sigma$-compact spaces can be found here.