Proposition 6

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

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

A connected topological group is generated by every neighbourhood of the identity element.

The converse of this proposition is in general false, as we shall see in Chapter IV (§ 2, no. 5). If a topological group $G$ is generated by each neighbourhood of the identity element, the most we can say is that $G$ contains no open subgroup other than $G$ .

\* As an example of a non-connected group $G$ which has an open subgroup distinct from $G$ , we may cite the multiplicative group $\mathbf{R}^*$ of non-zero real numbers, in which the subgroup $\mathbf{R}_+^*$ of real numbers $> 0$ is both open and closed (see Chapter IV, § 3, no. 2). \*

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

Cited by

Exercise 21, § 6

Cites

§ 2, no. 5
Chapter IV, § 3, no. 2