Non-Trivial Particular Point Topology is not T1
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Theorem
Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton.
Then $T$ is not a $T_1$ (Fréchet) space.
Proof
Follows directly from:
- Particular Point Topology is Closed Extension Topology of Discrete Topology
- Closed Extension Topology is not $T_1$.
$\blacksquare$
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: $4$