Corollary

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

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

Let G be a topological group, H a normal subgroup of G, and K a normal subgroup of G which contains H. Then the canonical bijection of G/K onto $(G/H)/(K/H)$ is an isomorphism of topological groups.

We know already that this bijection is an isomorphism of groups, and Proposition 22 (applied to the group K operating on the right on G) shows that it is a homeomorphism.

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

Cited by

Lemma 2, § 7
§ 1. Subgroups and quotient groups of $\mathbf{R}^n$

Cites

Proposition 22