Corollary 1

Tag 0CDT, permanent. Cite it as [Bourbaki, Tag 0CDT].

General Topology, chapter X, § 3. Special function spaces

Let $X$ be a locally compact space, $Y$ a topological space, $H$ a subset of $C(X; Y)$ . Then the compact-open topology on $H$ is the coarsest for which the mapping $(u, x) \to u(x)$ of $H \times X$ into $Y$ is continuous.

For, by Theorem 3, this mapping is continuous if and only if the canonical injection $H \to C_c(X; Y)$ is continuous.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Lemma 4, § 6
Proposition 3, § 9
Exercise 23, § 3
Exercise 25, § 3
Remark 3, § 3

Cites

Theorem 3