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

Statement

Let $X$ be a locally compact Hausdorff space, let $Y$ be a metric space, and let $K \subseteq C (X, Y)$ be compact in the compact-open topology. Then $K$ is equicontinuous.

Facts & Assumptions

Given: A locally compact Hausdorff space $X$ , a metric space $Y$ , and a compact compact-open family $K$ .

[L1]

Evaluation $C (X, Y) \times X \to Y$ is continuous for a locally compact Hausdorff domain (Evaluation is continuous for the compact-open topology on a locally compact Hausdorff domain).

[L2]

Compactness of a subspace may be used with open covers by ambient open sets (A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it).

[L3]

Equicontinuity requires one domain neighbourhood for every member of the family at the chosen point and tolerance (Equicontinuity on a topological domain and pointwise relative compactness).

Proof

technique · direct

1.1

If $X = \emptyset$ or $K = \emptyset$ , the conclusion is vacuous. Otherwise fix $x \in X$ and $ε > 0$ .

1.2

Call a pair $(O, U)$ of open sets admissible at $f \in K$ when $f \in O$ , $x \in U$ , and $d (g (y), f (x)) < ε / 3$ for every $g \in O$ and every $y \in U$ . Continuity of evaluation at $(f, x)$ , which [L1] supplies, makes at least one pair admissible at each $f \in K$ . Let $A$ be the set of all triples $(f, O, U)$ with $(O, U)$ admissible at $f$ . This set is defined outright and no pair is selected, so no choice principle is used.

2.1

The open sets $O$ occurring in triples of $A$ cover $K$ , because each $f \in K$ lies in the $O$ of some admissible triple. Applying [L2] to the cover indexed by $A$ yields finitely many triples $(f_{1}, O_{1}, U_{1}), \dots, (f_{m}, O_{m}, U_{m})$ of $A$ whose $O_{i}$ already cover $K$ ; only this finite selection is made. Put $U = U_{1} \cap \dots \cap U_{m}$ , an open neighbourhood of $x$ as a finite intersection.

L2step 1.2

3.1

Let $g \in K$ and take $i \leq m$ with $g \in O_{i}$ . If $y \in U$ then $y \in U_{i}$ and $x \in U_{i}$ , so admissibility at $f_{i}$ gives $d (g (y), f_{i} (x)) < ε / 3$ and $d (g (x), f_{i} (x)) < ε / 3$ , whence $d (g (y), g (x)) < 2 ε / 3 < ε$ . The one neighbourhood $U$ works for every $g \in K$ , which is what [L3] requires for equicontinuity at $x$ .

L3step 1.2step 2.1∎

Depends on

Used by

General Ascoli theorem for locally compact Hausdorff domains and metric targets Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 40 results over 14 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)