# Compact Hausdorff Topology is Maximally Compact

## Theorem

Let $T = \struct {S, \tau}$ be a Hausdorff space which is compact.

Then $\tau$ is maximally compact.

## Proof

Let $\tau'$ be a topology on $S$ such that $\tau \subseteq \tau'$ but that $\tau \ne \tau'$.

Consider the identity mapping $I_S: \struct {S, \tau'} \to \struct {S, \tau}$.

From Separation Properties Preserved in Subspace, $I_S$ is a continuous bijection from a Hausdorff space to a compact Hausdorff space.

Aiming for a contradiction, suppose $\struct {S, \tau'}$ is compact.

Then $I_S$ is also open, and so $\tau' \subseteq \tau$.

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Thus we have that $\tau \subseteq \tau'$ and $\tau' \subseteq \tau$.

Hence by definition of set equality:

- $\tau' = \tau$

But this contradicts our hypothesis that $\tau' \ne \tau$.

By Proof by Contradiction, it follows that no topology which is strictly finer than $\tau$ can be compact.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Compactness Properties and the $T_i$ Axioms