Remark 1

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

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

From condition b) of Proposition 24 it follows that every continuous homomorphism of a topological group into a discrete group is a strict morphism.

If G is compact and $f(G)$ is Hausdorff, then the bijective homomorphism $f$ associated with $f$ is bicontinuous (Chapter I, § 10, no. 2, Theorem 1, Corollary 2, and no. 1, Proposition 5, Corollary 4). Hence every continuous homomorphism of a compact group into a Hausdorff group is a strict morphism.

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

Cited by

§ 2. Continuous homomorphisms of $\mathbf{R}^n$ and its quotient groups

Cites

Corollary 2
and no. 1, Proposition 5
Corollary 4
Chapter I, § 10, no. 2, Theorem 1
Proposition 24