Lemma 1

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

General Topology, chapter III, § 7. Inverse limits of topological groups and rings

Under the hypotheses of Proposition 2, if $E$ is any closed subset of $G$ we have $E = \bigcap_\alpha E H_\alpha$ .

For $E$ is the intersection of the sets $EV$ where $V$ runs through the neighbourhood filter of $e$ [ $\S$ 3, no. 1, formula (1)], and every neighbourhood of $e$ contains an $H_\alpha$ ; whence the result, since $E \subset EH_\alpha$ .

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

Cited by

Proposition 3, § 7

Cites

§$ 3, no. 1
Proposition 2