Closure - Imported Knowlpedia Content

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

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

\overline{E}:=\bigcap\{\,F\subset X \mid F \text{ is closed and } E\subset F\,\}.

Equivalent characterizations

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

Remarks

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 $\overline{E}=X$ .