Excluded Point Space is not Perfectly T4

Theorem

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.

Then $T$ is not a perfectly $T_4$ space.

Proof

Excluded Point Topology is Open Extension Topology of Discrete Topology

Open Extension Topology is not Perfectly T4

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $13 \text { - } 15$. Excluded Point Topology: $4$