Accumulation Points of Sequence of Distinct Terms in Infinite Particular Point Space

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

Let $A \in \vartheta_p$ be a countably infinite open set of $T$.

Then $A$ 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 $\left \langle {x_n} \right \rangle$.