# Uncountable Particular Point Space is not Second-Countable

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## 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 10: \ 7$