Definition:Inversion Mapping/Topology

Theorem
Let $(G,\circ,\tau)$ be a topological group, then $\phi:G\to G$ such that $\forall x\in G$, $\phi(x)=x^{-1}$ is a homeomorphism.

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:
 * $\phi(\phi(x))=(x^{-1})^{-1}=x$

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.