Excluded Point Topology is T4/Proof 1

Theorem

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

Then $T$ is a $T_4$ space.

Proof

We have that an Excluded Point Space is Ultraconnected.

That means none of its closed sets are disjount.

Hence, vacuously, any two of its disjoint closed subsets of $S$ are separated by neighborhoods.

The result follows by definition of $T_4$ space.

$\blacksquare$

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: $2$