Closure

Let $(X,d)$ be a metric space and let $E\subset X$.

The closure of $E$, denoted $\overline{E}$, is defined as

Equivalently, $\overline{E}$ is the smallest closed set containing $E$.

A useful pointwise characterization is given by ball intersections , and in metric spaces there is also a sequence characterization (see closure via sequences ).

Examples:

• In $\mathbb{R}$, $\overline{(0,1)}=[0,1]$.
• If $E$ is closed, then $\overline{E}=E$.
• If $E$ is dense in $X$ (e.g., $\mathbb{Q}$ in $\mathbb{R}$), then $\overline{E}=X$.