Definition:Closure (Topology)/Definition 1

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Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$.


The closure of $H$ (in $T$) is defined as:

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

where $H'$ is the derived set of $H$.


That is, $H^-$ is the union of $H$ and its limit points.


Notation

The topological closure 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