A Topological space is Hausdorff, if for any pair of points $x, y \in X$ there exists open sets $O_1$ and $O_2$ such that $x \in O_1$, $y \in O_2$ and $O_1 \cap O_2 = \emptyset$ (have disjoint neighbourhoods). Also known as $T_2$ (see Separation axioms)

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Every finite set in a Hausdorff space is closed

In Hausdorff spaces, a convergent sequence has a unique Limit point (proof)

Another proposition saying that a Limit point of a set in a Hausdorff space intersects the set in infinitely many points. We need stronger condition to say that a limit point is the point to which a sequence converges. See more at Convergence