Sequential characterization of closure

In a metric space, a point lies in the closure of a set iff it is the limit of a sequence from the set.

Sequential characterization of closure

Sequential characterization of closure: Let $(X,d)$ be a metric space , let $A\subseteq X$ , and let $x\in X$ . Then

x\in \overline{A} \quad\Longleftrightarrow\quad \text{there exists a sequence }(a_n)\text{ in }A\text{ with }a_n\to x,

where $\overline{A}$ denotes the closure of $A$ .

This turns the topological notion of closure into a sequential condition in metric spaces, and it complements the sequential characterization of closed sets .