# Limit Point of Countable Open Set in Particular Point Space

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## Theorem

Let $T = \left({S, \tau_p}\right)$ be an infinite particular point space.

Let $U \in \tau_p$ be a countably infinite open set of $T$.

Then $U$ has a limit point.

## Proof

Let $\left \langle {a_i}\right \rangle$ be a convergent sequence in $T$ whose limit is $\alpha$.

Then by definition every open set in $T$ containing $\alpha$ contains all but a finite number of terms of $\left \langle {x_n} \right \rangle$.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 9 - 10: \ 1$