Definition:Sigma-Compact Space
(Redirected from Definition:Countable at Infinity)
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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
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): sigma-compact or $\sigma$-compact