The closure $\bar{A}$ of a set $A\subset X$ is the smallest Closed set in $X$ which contains $A$.

Let $(X,d)$ be a Metric Space which we know admits a Topological Space. Then the closure $\bar{A}$ of $A$ is given by: $$\bar{A}=A\cup \left\{ \lim_{ n \to \infty } a_{n}:a_{n}\in A \ ,\forall n\in\mathbb{N} \right\}$$