sigma-compact topological space (Rev #7, changes) in nLab

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Contents

Definition

A topological space is called sigma-compactσ\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 σ\sigma-compact space is also weakly locally compact?, then one can take the countable set of compact subspaces to be increasing, namely K i⊂K i+1K_i \subset K_{i+1}, and that K iK_i is in the interior of K i+1K_{i+1}.

Examples

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

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

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

• countable cover

References

• Wikipedia, σ-compact space