Proposition 16

Let G be a topological group and let H be a normal subgroup of G. Then the quotient by H of the topology of G is compatible with the group structure of G/H.

If x → x̂ is the canonical mapping of G onto G/H, then we have to show that (x̂, ŷ) → x̂ŷ^{-1} is a continuous mapping of (G/H) × (G/H) into G/H. Since the equivalence relation x^{-1}y ∈ H is open (no. 4, Lemma 2), it is enough to show that (x, y) → x̂ŷ^{-1} is a continuous mapping of G × G into G/H (Chapter I, § 5, no. 3, Corollary to Proposition 8, and § 3, no. 4, Proposition 6). But (x, y) → x̂ŷ^{-1} is the composition of the continuous mappings x → x̂ and (x, y) → xy^{-1}, hence is continuous.

Whenever in the sequel we consider a quotient group $G/H$ of a topological group $G$ as a topological group, it is always to be understood that the topology of $G/H$ is the quotient by $H$ of the topology of $G$, unless the contrary is expressly stated.