# Compact Hausdorff Space is T4

Jump to navigation
Jump to search

## Theorem

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

Then $T$ is a $T_4$ space.

## Proof

We have that a Compact Subspace of Hausdorff Space is Closed.

We also have that a Closed Subspace of Compact Space is Compact.

We also have that Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods.

$T$ is a $T_4$ space when any two disjoint closed subsets of $S$ are separated by neighborhoods.

Hence the result.

$\blacksquare$

## Sources

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