TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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General Ascoli theorem for locally compact Hausdorff domains and metric targets

Statement

Assume the Axiom of Choice. Let $X$ be a locally 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 locally compact Hausdorff space $X$ , a metric space $Y$ , and $F \subseteq C (X, Y)$ .

[L1]

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

[L2]

A compact compact-open family on a locally compact Hausdorff domain is equicontinuous (A compact compact-open family is equicontinuous on a locally compact Hausdorff domain).

[L3]

Every compact compact-open family is pointwise relatively compact (Every compact compact-open family is pointwise relatively compact).

[L4]

A closed subset of a compact 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

Suppose the compact-open closure ${\overline{F}}^{c o}$ is compact. By [L2] it is equicontinuous, and restricting its common neighbourhood estimates to $F$ shows that $F$ is equicontinuous.

1.2L3L4

By [L3], ${\overline{F}}^{c o}$ is pointwise relatively compact. For each $x$ , the closure of $F (x)$ is a closed subset of the compact closure of ${\overline{F}}^{c o} (x)$ , hence is compact by [L4]; thus $F$ is pointwise relatively compact.

1.3L1

Conversely, if $F$ is equicontinuous and pointwise relatively compact, [L1] says directly that its compact-open closure is compact.

2.1step 1.1step 1.2step 1.3∎

Steps 1.1--1.2 and 1.3 prove the two directions of the equivalence.

Depends on

Used by

Ascoli–Arzelà for a compact Hausdorff domain Corollary
Boundedness does not replace pointwise relative compactness for an arbitrary metric target Counterexample

Dependency tree · two levels

19 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)