CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Ascoli–Arzelà for a compact Hausdorff domain

Statement

Assume the Axiom of Choice. Let $X$ be a compact Hausdorff space, let $Y$ be a metric space, and let $F \subseteq C (X, Y)$ . The compact-open closure of $F$ is compact if and only if $F$ is equicontinuous and pointwise relatively compact.

Facts & Assumptions

Given: Choice, a compact Hausdorff space $X$ , a metric space $Y$ , and $F \subseteq C (X, Y)$ .

[L1]

The general Ascoli theorem gives the stated equivalence for locally compact Hausdorff domains (General Ascoli theorem for locally compact Hausdorff domains and metric targets).

[L2]

Every compact topological space is locally compact because the whole space is a compact neighbourhood of each point (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space).

Proof

technique · direct

1.1

By [L2], the compact space $X$ is locally compact; it is Hausdorff by hypothesis.

2.1

Apply [L1] to $X$ , $Y$ , and $F$ . This yields both directions of the claimed equivalence with the compact-open topology and the stated Choice hypothesis unchanged.

L1step 1.1∎

Depends on

Used by

Ascoli–Arzelà in the uniform topology for nonempty compact metric domains Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 79 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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