DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The compact-open topology on $C (X, Y)$ for arbitrary topological spaces

Definition

Let $X$ and $Y$ be topological spaces and let $C (X, Y)$ be the set of continuous maps from $X$ to $Y$ . For a compact subset $K \subseteq X$ and an open subset $V \subseteq Y$ , put

$S (K, V) : = {f \in C (X, Y) : f [K] \subseteq V} .$

The compact-open topology on $C (X, Y)$ is the topology generated by all sets $S (K, V)$ as a subbasis. In particular, $S (\emptyset, V) = C (X, Y)$ and $S (K, Y) = C (X, Y)$ , so the empty compact set and the whole target introduce no exceptional case.

Depends on

Dependency tree · 2 levels

Sources

Topology, second edition, Section 46

The compact-open topology on C(X,Y)C(X,Y) for arbitrary topological spaces

Definition

Depends on

Dependency tree · 2 levels

Sources

The compact-open topology on $C (X, Y)$ for arbitrary topological spaces