Definition:Closure (Metric Space)

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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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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.


Also see

  • Results about set closures can be found here.


Sources