Non-Trivial Particular Point Topology is not T4/Mistake

Source Work

 * Part $\text {II}$: Counterexamples
 * Section $8 \text { - } 10$: Particular Point Topology
 * Item $4$
 * Item $4$

Mistake

 * Every particular point topology is $T_0$, but since there are no disjoint open sets, none of the higher separation axioms are satisfied unless $X$ has only one point.

In the above, $X$ is a particular point space.

However, this is not true for the $T_4$ axiom.

The Sierpiński space is a particular point topology with exactly two points.

But the Sierpiński Space is T4.

Also see

 * Particular Point Topology with Three Points is not T4