Continuous Bijection from Compact to Hausdorff is Homeomorphism

Theorem
Let $T_1$ be a compact space.

Let $T_2$ be a Hausdorff space.

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

Then $f$ is a homeomorphism.

Proof
Let $g = f^{-1}$.

We need to show that $g: T_2 \to T_1$ is continuous.

For any $V \subseteq T_1$, we have $g^{-1} \sqbrk V = \map f V$.

We are to show that if $V$ is closed in $T_1$, then $g^{-1} \sqbrk V$ is closed in $T_2$.

Suppose $V$ is closed in $T_1$.

Since $T_1$ is compact, $V$ is compact by Closed Subspace of Compact Space is Compact.

So $f \sqbrk V$ is compact from Continuous Image of Compact Space is Compact.

Since $T_2$ is Hausdorff, $f \sqbrk V$ closed by Compact Subspace of Hausdorff Space is Closed.

But $f \sqbrk V = g^{-1} \sqbrk V$, so $g^{-1} \sqbrk V$ is closed.

From Continuity Defined from Closed Sets, it follows that $g$ is continuous.

Thus by definition, $f$ is a homeomorphism.