# 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:

$\blacksquare$