CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-16

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Under the Axiom of Choice, for a nonempty compact metric domain $X$ and a proper metric target $Y$ , the subsets of $C (X, Y)$ compact in the uniform topology are exactly the families closed in that topology that are pointwise bounded and equicontinuous

Statement

Assume the Axiom of Choice. Let $X$ be a nonempty compact metric space and let $Y$ be a proper metric space, meaning that every closed bounded subset of $Y$ is compact. A family $F \subseteq C (X, Y)$ is compact in the uniform topology if and only if it is closed in that topology, equicontinuous, and pointwise bounded, where pointwise bounded means that $F (x)$ is a bounded subset of $Y$ for every $x \in X$ ; the empty subset is bounded.

Facts & Assumptions

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

[L1]

The uniform closure of a family is compact exactly when the family is equicontinuous and every coordinate set has compact closure (Ascoli–Arzelà in the uniform topology for nonempty compact metric domains).

[L2]

A compact subset of a metric space is closed and bounded (A compact subset of a metric space is closed and bounded).

Proof

technique · direct

1.1L2

Suppose $F$ is compact in the uniform topology. Since this topology is metrizable, [L2] makes $F$ closed; hence its uniform closure is itself.

1.2given

Conversely suppose $F$ is uniformly closed, equicontinuous, and pointwise bounded. For each $x$ , the closure $\overline{F (x)}$ is closed and remains bounded; this also holds when $F (x) = \emptyset$ .

2.1L1L2step 1.1

By [L1], $F$ is equicontinuous and each $\overline{F (x)}$ is compact. By [L2] each such coordinate closure is bounded, so $F$ is pointwise bounded.

2.2L1step 1.2

The coordinate closure $\overline{F (x)}$ is closed and bounded, hence compact by properness of $Y$ . Thus [L1] makes the uniform closure of $F$ compact.

3.1step 1.1step 2.1step 2.2∎

Since $F$ is uniformly closed, it equals that compact closure and is compact. Steps 1.1--1.2 prove the converse implication, completing the equivalence.

Depends on

Used by

Real and finite-dimensional Euclidean Ascoli–Arzelà criteria Corollary

Dependency tree · two levels

17 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

The Ascoli–Arzelà Theorem, BBT (standard reference, not scraped)
Topology, second edition, Corollary 45.5 (standard reference, not scraped)

Under the Axiom of Choice, for a nonempty compact metric domain XX and a proper metric target YY, the subsets of C(X,Y)C(X,Y) compact in the uniform topology are exactly the families closed in that topology that are pointwise bounded and equicontinuous