Inverse Mapping in Topological Group is Homeomorphism
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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 the group. Then, from Inverse of an Inverse applied to the group structure $\phi(\phi(x))=(x^{-1})^{-1}=x$. Hence $\phi\circ\phi=Id_G$ and, 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.
$\blacksquare$
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