Excluded Point Topology is T4

Theorem

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

Then $T$ is a $T_4$ space.

Proof 1

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.

$\blacksquare$

Proof 2

We have:

Excluded Point Topology is Open Extension Topology of Discrete Topology

Open Extension Topology is $T_4$

Hence the result.

$\blacksquare$

Proof 3

We have:

Excluded Point Topology is $T_5$

$T_5$ Space is $T_4$

$\blacksquare$