CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28

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The compact-open topology on C(Ω,C) is independent of the chosen compact exhaustion

Statement

Any two compact exhaustions of a plane domain $Ω$ induce the same topology on $C (Ω, C)$ by the weighted exhaustion metric. Equivalently, the compact-open topology on $C (Ω, C)$ is independent of the chosen compact exhaustion.

Facts & Assumptions

Given: Two compact exhaustions of the same plane domain $Ω$ .

[L1]

An exhaustion metric induces exactly local uniform convergence (The exhaustion metric induces exactly the topology of locally uniform convergence).

[L2]

On a metric domain and metric target, the compact-open topology is the topology of compact convergence (For a metric domain and a metric target the compact-open topology on $C (X, Y)$ is the topology of compact convergence).

Proof

technique · direct

1.1

By [L1], each exhaustion metric induces the same convergence notion, namely local uniform convergence on $Ω$ .

L1given

2.1

Fact [L2] identifies that convergence notion with compact convergence and hence with the compact-open topology on $C (Ω, C)$ . Therefore the induced topology is independent of the chosen exhaustion.

L2given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

30 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

Matthias Weber, Complex Analysis, Ch. 5 §§5.1-5.2 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 2 §5.2 and Ch. 8 §3.2 (standard reference, not scraped)
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, Ch. 2 (standard reference, not scraped)