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

Theorem
Let $T = \struct {S, \tau_p}$ 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 $\sequence {a_i}$ 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 $\sequence {a_i}$.