Inversion Mapping on Topological Group is Homeomorphism

Theorem
Let $T = \left({G, \circ, \tau}\right)$ be a topological group.

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

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

Let $x \in G$ be any element of $G$.

From Inverse of Group Inverse applied to the group structure:
 * $\phi \left({\phi \left({x}\right)}\right) = \left({x^{-1}}\right)^{-1} = x$

Hence:
 * $\phi \circ \phi = Id_G$

In particular, $\phi$ is bijective from Bijection iff Left and Right Cancellable.

$\phi$ is its own inverse, and thus $\phi$ is continuous, bijective and its inverse (also $\phi$) is continuous; the definition of homeomorphism.