Definition:Closure (Topology)/Definition 5

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

The closure of $H$ (in $T$) is the union of the set of all isolated points of $H$ and the set of all limit points of $H$:

$H^- := H^i \cup H'$

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.