Proposition 2

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

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

If $G$ is a Hausdorff topological group, then the closure of a commutative subgroup of $G$ is a commutative subgroup of $G$ .

By reason of Proposition 1, we may limit ourselves to the case where $H$ is dense in $G$ . The continuous functions $xy$ and $yx$ are equal on $H \times H$ , and are therefore equal on $G \times G$ , by virtue of the principle of extension of identities (Chapter I, § 8, no. 1, Proposition 2, Corollary 1).

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

Cited by

Nothing.

Cites

Chapter I, § 8, no. 1, Proposition 2
Corollary 1
Proposition 1