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.

Corollary
Let $$T_1$$ be a compact space.

Let $$T_2$$ be a Hausdorff space.

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

Then $$f$$ determines a homeomorphism from $$T_1$$ to $$f \left({T_1}\right)$$.

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 $$g^{-1} \left({V}\right) = f \left({V}\right)$$.

We are to show that if $$V$$ is closed in $$T_1$$, then $$g^{-1} \left({V}\right)$$ 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 $$f \left({V}\right)$$ is compact from Continuous Image of a Compact Space is Compact.

Since $$T_2$$ is Hausdorff, $$f \left({V}\right)$$ closed by Compact Subspace of Hausdorff Space is Closed.

But $$f \left({V}\right) = g^{-1} \left({V}\right)$$, so $$g^{-1} \left({V}\right)$$ is closed.

From Continuity Defined by Closure, it follows that $$g$$ is continuous.

Thus by definition, $$f$$ is a homeomorphism.

Proof of Corollary
Follows from the above and Continuity of Composite with Inclusion.