# Definition:Closure (Metric Space)

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## Contents

## 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

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.6$: Open Sets and Closed Sets: Exercise $6$