Compact Hausdorff Space is T4
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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