CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Under Choice, an equicontinuous family into a compact metric target has compact compact-open closure

Statement

Assume the Axiom of Choice. Let $X$ be any topological space, let $Y$ be a compact metric space, and let $F \subseteq C (X, Y)$ be equicontinuous. Then the compact-open closure of $F$ is compact.

Facts & Assumptions

Given: Choice, a topological space $X$ , a compact metric space $Y$ , and an equicontinuous family $F \subseteq C (X, Y)$ .

[L1]

Under Choice, equicontinuity and pointwise relative compactness give compact compact-open closure (Under Choice, equicontinuity and pointwise relative compactness give compact compact-open closure).

[L2]

Every closed subset of a compact topological space is compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact).

Proof

technique · direct

1.1L2

For each $x \in X$ , the closure $\overline{F (x)}$ is a closed subset of the compact target $Y$ , and is therefore compact by [L2]. This includes the empty coordinate set.

2.1L1step 1.1∎

Thus $F$ is pointwise relatively compact. Together with the assumed equicontinuity, [L1] makes its compact-open closure compact. No local compactness hypothesis on $X$ is used.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Topology, second edition, Theorem 47.1 (standard reference, not scraped)