Inverse of Homeomorphism is Homeomorphism

Theorem

Let $T, T'$ be topological spaces.

Let $f: T \to T'$ be a homeomorphism.

Then $f^{-1}: T' \to T$ is also a homeomorphism.

Proof

By definition, a homeomorphism is a bijection such that both $f$ and $f^{-1}$ are continuous.

As $f$ is a bijection then by Bijection iff Inverse is Bijection, so is $f^{-1}$.

So by definition $f^{-1}$ is a bijection such that both $f^{-1}$ and $\left({f^{-1}}\right)^{-1}$ are continuous.

The result follows from Inverse of Inverse of Bijection.

$\blacksquare$