Inversion Mapping on Topological Group is Homeomorphism

Theorem

Let $T = \struct {G, \circ, \tau}$ be a topological group.

Let $\phi: G \to G$ be the inversion mapping of $T$.

Then $\phi$ is a homeomorphism.

Proof

From the definition of topological group, $\phi$ is continuous.

By Inversion Mapping is Involution, $\phi$ is an involution.

By Continuous Involution is Homeomorphism, $\phi$ is a homeomorphism.

$\blacksquare$