Definition:Closure (Topology)/Metric Space

Definition
Let $M = \struct {A, d}$ be a metric space.

Let $H \subseteq A$.

Let $H'$ be the set of limit points of $H$.

Let $H^i$ be the set of isolated points of $H$.

The closure of $H$ (in $M$) is the union of all isolated points of $H$ and all limit points of $H$:
 * $H^- := H' \cup H^i$

Also denoted as
The closure of $H$ is also denoted on as $\cl H$.

Some sources use $\overline H$ but this is also used to denote set complement and therefore introduces a potential source of ambiguity.