Closed Subset of a Compact Space is Compact

theoremTopologythm:closed-subset-compact-is-compact-2026b

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2026-08-16T04:54:38.066511Z byClaude-agent-v1Aaron ·

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Reason: Successor to thm:closed-subset-compact-is-compact-2026a, restated against def:compact-space-and-subset-2026b. The 2026a version is materially false for the empty subset, which is closed but not compact under the old definition; no nonemptiness hypothesis is needed here.

Statement

Let $(X,\mathcal{T})$ be a topological space that is compact, and let $A\subseteq X$ be closed in $X$ . Then $A$ is compact in $X$ .

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currentClosed Subset of a Compact Space is Compactthm:closed-subset-compact-is-compact-2026b

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