Limit Points in T1 Space

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space which satisfies the $T_1$ (Fréchet) axiom.

Let $H \subset S$ be any subset of $S$.

Let $x \in H$.

Then $x$ is a limit point of $H$ iff every neighborhood of $x$ contains infinitely many points of $H$.