nLab sigma-compact topological space

Definition

A topological space is called $\sigma$-compact if it is the union of a countable set of compact subspaces.

Properties

• locally compact and sigma-compact spaces are paracompact

If a $\sigma$-compact space is also weakly locally compact?, then one can take the countable set of compact subspaces to be increasing, namely $K_i \subset K_{i+1}$, and that $K_i$ is in the interior of $K_{i+1}$.

Examples

Every compact space is trivially compact. A discrete space is $\sigma$-compact if and only if it is countable. The product of a finite number of $\sigma$-compact spaces is $\sigma$-compact.

• locally compact and second-countable spaces are sigma-compact

This includes $\mathbb{R}^n$ with the Euclidean topology.

related concepts

• countable cover

References

• Wikipedia, σ-compact space