Proposition 3

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

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

Let $G$ be a Hausdorff topological group and let $M$ be any subset of $G$ . Then the set $M'$ of elements of $G$ which commute with each element of $M$ is a closed subgroup of $G$ . In particular, the centre of $G$ is closed in $G$ .

For $M'$ is the intersection of the sets $F_m (m \in M)$ , where $F_m$ is the set of all $x \in G$ such that $xm = mx$ ; and $F_m$ is closed (Chapter I, § 8, no. 1, Proposition 2).

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