Non-Trivial Excluded Point Topology is not T1

Theorem

Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space such that $S$ is not a singleton.

Then $T$ is not a $T_1$ (Fréchet) space.

Proof

Follows directly from:

Excluded Point Topology is Open Extension Topology of Discrete Topology

Open Extension Topology is not $T_1$

$\blacksquare$

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: $2$