# Non-Trivial Particular Point Topology is not T4

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## Source Work

1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: *Counterexamples in Topology*:

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

- Section $8 - 10$: Particular Point Topology

This mistake can be seen in the second edition (1978) as republished by Dover in 1995: ISBN 0-486-68735-X

## 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.