Bijection is Open iff Inverse is Continuous

Theorem
Let $T_1 = \left({X_1, \vartheta_1}\right), T_2 = \left({X_2, \vartheta_2}\right)$ be topological spaces.

Let $f: T_1 \to T_2$ be a bijection.

Then $f$ is open iff $f^{-1}$ is continuous.

Proof
Let $f$ be a bijection and let $g := f^{-1}$.

By Bijection iff Inverse is Bijection we have that $g$ is a bijection and that $g^{-1} = f$.

Let $f$ be open.

Then by definition of open mapping:
 * $\forall H \in \vartheta_1: f \left({H}\right) \in \vartheta_2$

taking $H \in \vartheta_1$ by definition of open in $T_1$.

But $f = g^{-1}$ and so:


 * $\forall H \in \vartheta_1: g^{-1} \left({H}\right) \in \vartheta_2$

which is exactly the definition for $g$ to be continuous.

The argument works the other way.

Let $g$ be continuous.

Then by definition of open mapping:
 * $\forall H \in \vartheta_1: g^{-1} \left({H}\right) \in \vartheta_2$

But $g^{-1} = f$ and so:
 * $\forall H \in \vartheta_1: f \left({H}\right) \in \vartheta_2$

which is exactly the definition for $g$ to be open.