Continuous Bijection from Compact to Hausdorff is Homeomorphism

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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 $f \left({T_1}\right)$.

That is, $f$ is an embedding of $T_1$ into $T_2$.


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

Thus by definition, $f$ is a homeomorphism.

$\blacksquare$


Sources

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