Convergence
Suppose that $$\left( \mathcal{ T }, X \right)$$ is a topological space, then a sequence of points $$\left( x_n \right)$$ in $$X$$ is said to converge to the point $$c \in X$$ provided that for each neighborhood of $$c$$, there exists some $$N \in \mathbb { N }_1$$ such that for all $$n \ge N$$, we have $$x_n \in U$$
Hausdorff Space
A topological space $$X$$ is called a Hausdorff space if for each pair of points $$x_1, x_2$$ of distinct points of $$X$$, there exist neighborhoods $$U_1, U_2$$ of $$x_1, x_2$$ respectively that are disjoint
Every one point set is Closed in a Hausdorff Space
Suppose that $$p \in X$$ where $$X$$ is a Hausdorff space, then $$\left\{ p \right\}$$ is closed

We will show that $$\left\{ p \right\}$$ is closed by showing $$\left\{ p \right\} = \overline{ \left\{ p \right\} }$$

Suppose that $$y \in X$$ and $$y \neq p$$, since $$X$$ was Hausdorff, we get $$U_p, U_y$$ disjoint neighborhoods. Since $$U_y \cap U_p = \emptyset$$ , since $$U_p \supseteq \left\{ p \right\}$$, then $$U_y \cap \left\{ p \right\} = \emptyset$$ therefore $$y \notin \overline{ \left\{ p \right\} }$$.

Since $$\left\{ p \right\} \subseteq \overline{ \left\{ p \right\} }$$, then we know $$p \in \overline{ \left\{ p \right\} }$$, so therefore $$\overline{ \left\{ p \right\} } = \left\{ p \right\}$$ as needed.

Every Finite Set is Closed in a Hausdorff Space