Excluded Point Topology is T4

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Theorem

Let $T = \left({S, \tau_{\bar p}}\right)$ 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 $T_4$

$\blacksquare$