Inverse of Homeomorphism between Metric Spaces is Homeomorphism

Theorem
Let $M = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}$ be metric spaces.

Let $f: M_1 \to M_2$ be a homeomorphism.

Then $f^{-1}: M_2 \to M_1$ 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 $\paren {f^{-1} }^{-1}$ are continuous.

The result follows from Inverse of Inverse of Bijection.