Uncountable Particular Point Space is not Second-Countable

Theorem

Let $T = \struct {S, \tau_p}$ be an uncountable particular point space.


Then $T$ is not second-countable.


Proof

Let $H = S \setminus \set p$ where $\setminus$ denotes set difference.

Every subset $V \subseteq H$ is a closed set from Subset of Particular Point Space is either Open or Closed.

Thus we can consider $H$ as an uncountable discrete space.

The result follows from Uncountable Discrete Space is not Second-Countable.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $10$. Uncountable Particular Point Topology: $7$
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