Proposition 4

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

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

Suppose that $G$ is Hausdorff, the $H_\alpha$ are compact and that (AP) is satisfied. If $C_\alpha$ is the identity component of $G_\alpha = G/H_\alpha$ , then the identity component $C$ of $G$ can be identified with $\varprojlim C_\alpha$ , and we have $f_\alpha(C) = C_\alpha$ .

The proposition is a consequence of the following lemma:

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

Cited by

Lemma 1, Appendix 1

Cites

Nothing.