Proposition 17

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

General Topology, chapter III, § 2. Subgroups, quotient groups, homomorphisms homogeneous spaces, product groups

Let $\varphi$ be the canonical mapping of a topological group $G$ onto a quotient group $G/H$ . If $\mathfrak{B}$ is a fundamental system of neighbourhoods of $e$ in $G$ , then $\varphi(\mathfrak{B})$ is a fundamental system of neighbourhoods of the identity element $\varphi(e)$ of $G/H$ .

This is a particular case of Proposition 5 of Chapter I, § 5, no. 3.

Propositions 13 and 14 give in particular, for quotient groups:

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

Cited by

Proposition 19, § 2
§ 3. Uniform structures on groups
Proposition 4, § 3

Cites

Proposition 5 of Chapter I, § 5, no. 3