Closure

The smallest closed set containing a given set, equivalently points arbitrarily close to it.

Closure

Let $(X,d)$ be a metric space and let $A\subseteq X$ . The closure of $A$ , denoted $\overline{A}$ (or $\operatorname{cl}(A)$ ), is the set

\overline{A}:=\{x\in X : \forall r>0,\ B(x,r)\cap A\neq\varnothing\}.

Equivalently, $\overline{A}$ is the intersection of all closed sets that contain $A$ .

The closure adds to $A$ all points that can be approximated arbitrarily well by points of $A$ . It is fundamental for density , limit points , and topological convergence.

Examples:

In $\mathbb{R}$ , $\overline{(0,1)}=[0,1]$ .
In $\mathbb{R}$ , $\overline{\mathbb{Q}}=\mathbb{R}$ (rationals are dense).
If $A$ is closed, then $\overline{A}=A$ .