Proposition 2

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

General Topology, chapter III, § 1. Topologies on groups

A topological group $G$ is Hausdorff if and only if the set $\{e\}$ is closed.

Clearly, if $G$ is Hausdorff, then $\{e\}$ is closed. Conversely, if $\{e\}$ is closed, then the diagonal $\Delta$ of $G \times G$ is closed, because it is the inverse image of $\{e\}$ under the continuous mapping $(x, y) \to xy^{-1}$ ; hence (Chapter I, § 8, no. 1, Proposition 1) $G$ is Hausdorff.

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

Cited by

Example, § 1
Proposition 1, § 2
Example 2, § 6

Cites

Chapter I, § 8, no. 1, Proposition 1