Continuous Involution is Homeomorphism

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.