Compact Space is Sigma-Compact

Theorem

Every compact space is $\sigma$-compact.


Proof

By definition, a $\sigma$-compact space is the union of countably many compact sets.

A compact space is the union of exactly one compact space.

Hence the result.

$\blacksquare$


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
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