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.

$\blacksquare$