Definition:Closure (Topology)/Metric Space
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
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets: Exercise $6$