Definition:Closure (Topology)/Definition 4

From ProofWiki
Jump to navigation Jump to search

Definition

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

Let $H \subseteq S$.


The closure of $H$ (in $T$) is defined as the union of $H$ and its boundary in $T$:

$H^- := H \cup \partial H$


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.