Proposition 8

Tag 0BI1, permanent. Cite it as [Bourbaki, Tag 0BI1].

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

Let $H$ be a dense subgroup of a topological group $G$ , and let $K$ be a normal subgroup of $H$ . Then the closure $\overline{K}$ of $K$ in $G$ is a normal subgroup of $G$ .

For the mapping $(z, x) \to zxz^{-1}$ is continuous on $G \times G$ , and maps $H \times K$ into $K$ ; hence (Chapter I, § 2, no. 1, Theorem 1) it maps $G \times \overline{K} = \overline{H} \times K$ into $\overline{K}$ .

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

Cited by

Exercise 8, § 2
Proposition 5, § 6
Corollary 1, § 3

Cites

Chapter I, § 2, no. 1, Theorem 1