Continuous Involution is Homeomorphism
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Theorem
Let $\struct {S, \tau}$ be a topological space.
Let $f: S \to S$ be a continuous involution.
Then $f$ is a homeomorphism.
Proof
From Involution is Permutation, $f$ is a permutation and so a bijection.
Since $f$ is continuous, it suffices to verify that its inverse is also continuous.
Now recall $f$ is an involution, that is, $f^{-1} = f$.
Thus $f^{-1}$ is also continuous.
Hence $f$ is a homeomorphism.
$\blacksquare$