Corollary 3

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

General Topology, chapter II, § 1. Uniform spaces

Every uniform space satisfies axiom $(O_{III})$ .

If $x$ is any point of $X$ and $V$ runs through the entourages of $X$ which are closed in $X \times X$ , then the sets $V(x)$ form a fundamental system of neighbourhoods of $x$ in $X$ by Corollary 2, and they are closed in $X$ (Chapter I, , no. 2, Corollary to Proposition 4)

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

Cited by

Proposition 3, § 1
Definition 1, § 3

Cites

Corollary 2