Definition:Closure (Topology)/Definition 6
Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
The closure of $H$ (in $T$), denoted $H^-$, is the set of all adherent points of $H$.
Adherent Point
Let $A \subseteq S$.
A point $x \in S$ is an adherent point of $H$ if and only if $x$ is an element of the closure of $H$.
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.
Also see
- Results about set closures can be found here.
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (next): $\S 4.6$: Closure, Interior, Boundary: Definition $4.2$