Definition:Closure (Topology)/Metric Space

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This page is about Closure in the context of Metric Spaces. For other uses, see Closure.

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 $H$:

$H^- := H' \cup H^i$


Notation

The closure operator of $H$ is variously denoted:

$\map \cl H$
$\map {\mathrm {Cl} } H$
$\overline H$
$H^-$


Of these, it can be argued that $\overline H$ has more problems with ambiguity than the others, as it is also frequently used for the set complement.

$\map \cl H$ and $\map {\mathrm {Cl} } H$ are cumbersome, but they have the advantage of being clear.

$H^-$ is neat and compact, but has the disadvantage of being relatively obscure.


On $\mathsf{Pr} \infty \mathsf{fWiki}$, $H^-$ is notation of choice, although $\map \cl H$ can also be found in places.


Examples

Union of Disjoint Closed Real Intervals

Let $\R$ be the real number line under the usual (Euclidean) metric.

Let $M$ be the subspace of $\R$ defined as:

$M = \closedint 0 1 \cup \closedint 2 3$

Let $\map {B_1} 1$ denote the open $1$-ball of $1$ in $M$.

Let $\map { {B_1}^-} 1$ denote the closed $1$-ball of $1$ in $M$.

Then:

$\map \cl {\map {B_1} 1} = \closedint 0 1$

while:

$\map { {B_1}^-} 1 = \closedint 0 1 \cup \set 2$


Also see

  • Results about set closures can be found here.


Sources

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