Non-Trivial Particular Point Topology is not T4/Mistake

Source Work

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.):

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


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $8 \text { - } 10$. Particular Point Topology: $4$
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