Proposition 1

Tag 0CD8, permanent. Cite it as [Bourbaki, Tag 0CD8].

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

If $Y$ is a metrizable uniform space, the uniformity of uniform convergence on $\mathscr{F}(X; Y)$ is metrizable.

For if $d$ is a metric on $Y$ compatible with its uniform structure, the structure of uniform convergence on $\mathscr{F}(X; Y)$ is defined by the single pseudometric

\varepsilon(u, v) = \sup_{x \in X} d(u(x), v(x));

in general this pseudometric is not finite, but it is equivalent to a finite one (Chapter IX, § 1, no. 2), and since the uniformity of uniform convergence is Hausdorff (§ 1, no. 2, Proposition 1), it is metrizable.

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

Cited by

Nothing.

Cites

Chapter IX, § 1, no. 2
§ 1, no. 2, Proposition 1