Corollary 1

Tag 0CB5, permanent. Cite it as [Bourbaki, Tag 0CB5].

General Topology, chapter X, § 1. The uniformity of $\mathscr{G}$-convergence

A subspace $H$ of $\mathscr{F}_\mathscr{S}(X; Y)$ is complete if and only if, for each Cauchy filter $\Phi$ on $H$ , there exists $u_0 \in H$ such that $\Phi$ converges pointwise to $u_0$ in $B = \bigcup_{A \in \mathscr{S}} A$ .

This follows immediately from Proposition 5.

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

Cited by

Corollary 2, § 1

Cites

Proposition 5