Proposition 1

Tag 0A78, permanent. Cite it as [Bourbaki, Tag 0A78].

Lie Groups and Lie Algebras, chapter III, § 1. Lie groups

A Lie group is a complete metrizable topological group.

Since $e$ admits an open neighbourhood homeomorphic to an open ball of a normed space, $e$ admits a countable fundamental system of neighbourhoods whose intersection is $\{e\}$ . Hence $G$ is metrizable (General Topology, Chapter III, § 1, Corollary to Proposition 2 and Chapter IX, § 3, Proposition 1). By Lemma 1, there exists a neighbourhood of $e$ which is complete under the right uniform structure and hence $G$ is complete (General Topology, Chapter III, § 3, Proposition 4).

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

Cited by

Proposition 2, § 1
Proposition 2, § 6

Cites

Lemma 1
Chapter IX, § 3, Proposition 1
*General Topology*, Chapter III, § 1, Corollary to Proposition 2
*General Topology*, Chapter III, § 3, Proposition 4