Separation Properties in Open Extension of Particular Point Topology

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Theorem

Let $T = \left({S, \tau_p}\right)$ be a particular point space such that $S$ is not a singleton or a doubleton.

Let $T^*_{\bar q} = \left({S^*_q, \tau_{\bar q}}\right)$ be an open extension space of $T$.


Then:

$T^*_{\bar q}$ is a $T_0$ (Kolmogorov) space.
$T^*_{\bar q}$ is a $T_4$ (space.
$T^*_{\bar q}$ is not a $T_1$ (Fréchet) space.
$T^*_{\bar q}$ is not a $T_5$ (space.


Proof

We have that a Particular Point Space is $T_0$.

Then from Condition for Open Extension Space to be $T_0$ Space, it follows that $T^*_{\bar q}$ is a $T_0$ (Kolmogorov) space.


We have directly that:

An Open Extension Topology is not $T_1$.
An Open Extension Topology is $T_4$.


Finally, we have that a Particular Point Topology with three points or more is not $T_4$.

From $T_5$ Space is $T_4$ Space, it follows that $T$ is not a $T_5$ space.

It follows from Condition for Open Extension Space to be $T_5$ Space that $T^*_{\bar q}$ is not a $T_5$ space.

$\blacksquare$


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