Universal Property of Quotient of Topological Group

Theorem

Let $G$ and $H$ be topological groups.

Let $N$ be a normal subgroup of $G$.

Let $\pi : G \to G/N$ be the quotient mapping.

Let $f : G \to H$ be a continuous group homomorphism whose kernel contains $N$.

Then there exists a unique continuous group homomorphism $\overline f: G / N \to H$ such that $f = \overline f \circ \pi$.

Proof

Because $N \subset \ker f$, $f$ is constant on the cosets of $N$.

Thus $f$ is invariant under congruence modulo $N$.

By Universal Property of Quotient Set, there exists a unique mapping $\overline f: G / N \to H$ such that $f = \overline f \circ \pi$.

It suffices to verify that it is a continuous group homomorphism.

By Universal Property of Quotient Space, there exists a continuous mapping $\overline g: G / N \to H$ such that $f = \overline g \circ \pi$.

By uniqueness of $f$, $g = f$.

By Universal Property of Quotient Group, there exists a group homomorphism $\overline h: G / N \to H$ such that $f = \overline h \circ \pi$.

By uniqueness of $f$, $h = f$.

$\blacksquare$