Closure (Metric Space)

Definition

Let $(X,d)$ be a metric space and $A\subseteq X$. The closure of $A$, denoted $\overline{A}$, is the set of all adherent points of $A$: $\overline{A}=\{x\in X\mid \forall \epsilon >0,B(x,\epsilon )\cap A\ne \varnothing \}$

Properties

1. $\overline{A}$ is the smallest closed set containing $A$.
2. $A$ is closed iff $A=\overline{A}$.
3. $\overline{A}=A\cup L$, where $L$ is the set of all limit points of $A$.