Equivalence of Definitions of Homeomorphic Metric Spaces

Theorem
Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces.

Let $f: T_\alpha \to T_\beta$ be a bijection.

The following definitions for $f$ to be a homeomorphism are equivalent:

Proof
In order to prove the assertion it is sufficient to prove that the conditions for homeomorphism in definitions $2$ to $4$ are necessary and sufficient conditions for $f$ and $f^{-1}$ to be continuous on $M_1$ and $M_2$ respectively.

Definition 1 implies Definition 2
Let $f$ and $f^{-1}$ both be continuous by definition $2$.

Let $U \subseteq A_1$ be open in $M_1$.

As $f^{-1}$ is continuous, $\left({f^{-1}}\right)^{-1} \left[{U}\right] = f \left[{U}\right]$ is open in $M_2$.

That is, $f \left[{U}\right] = f \left[{U}\right]$ is open in $M_2$.

Let $f \left[{U}\right] \subseteq A_2$ be open in $M_2$.

Then as $f$ is continuous, $f^{-1} \left[{f \left[{U}\right]}\right] = U$ is open in $M_1$.

Thus:
 * for all $U \subseteq A_1$, $U$ is an open set of $M_1$ $f \left[{U}\right]$ is an open set of $M_2$.