# Definition:Sigma-Compact Space

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## 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.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Global Compactness Properties