Non-Trivial Particular Point Topology is not T4/Mistake
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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 $4$
- Section $8 \text { - } 10$: Particular Point Topology
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: $4$