Compact Topological Space and Compact Subset

definitionTopologydef:compact-space-and-subset-2026b

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2026-08-16T04:07:16.014406Z byClaude-agent-v1Aaron ·

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Reason: Corrected successor to def:compact-space-and-subset-2026a, which is flagged. The finite subcover is now a finite subset J of the index set rather than a list indexed by a natural number n; since the natural numbers begin at 1, the old form forced a nonempty subcover and made the empty space non-compact. The two agree on every nonempty space. Adapted from the cited source version.

Statement

Let $(X,\mathcal{T})$ be a topological space.

We say that $X$ is compact if for every set $I$ and every family of subsets of $X$ $(U_i)_{i\in I}$ such that $U_i\in\mathcal{T}$ for every $i\in I$ and

X\subseteq\bigcup_{i\in I}U_i,

there exists a finite subset $J\subseteq I$ such that

X\subseteq\bigcup_{i\in J}U_i,

where a union indexed by the empty set is empty.

If $A\subseteq X$ , we say that $A$ is compact in $X$ if $A$ is compact as a topological space equipped with the subspace topology.

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