The closure $\overline{A}$ of a subset $A$ of a topological space $X$ is the intersection of all closed sets containing $A$

Equivalently, $\overline{A}$ consists of $A$ together with all limit points of $A$ in $X$ or equivalently $x\in\overline{A}$ if and only if every neighborhood of $x$ intersects $A$ Sometimes the notation $\operatorname{cl}(A)$ is used.

If it is not clear, which topological space is used, one writes $\overline{A}^X$ Note that if $Y$ is a subspace of $X$ then $\overline{A}^X$ may not be the same as $\overline{A}^Y$ For example, if $X=\mathbb{R}$ $Y=(0,1)$ and $A=(0,1)$ then $\overline{A}^X=[0,1]$ while $\overline{A}^Y=(0,1)$