Separation Properties in Open Extension of Particular Point Topology

Theorem

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

Let $T^*_{\bar q} = \struct {S^*_q, \tau^*_{\bar q} }$ 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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $16$. Open Extension Topology: $10$