Definition:Closure (Topology)

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Definition

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

Let $H \subseteq S$.


Definition 1

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

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

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


Definition 2

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

$\displaystyle H^- := \bigcap \left\{{K \supseteq H: K}\right.$ is closed in $\left.{T}\right\}$


Definition 3

The closure of $H$ (in $T$), denoted $H^-$, is defined as the smallest closed set of $T$ that contains $H$.


Definition 4

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

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


Definition 5

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 $H$:

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


Notation

The topological closure of $H$ is variously denoted:

  • $\operatorname{cl} \left({H}\right)$
  • $\operatorname{Cl} \left({H}\right)$
  • $\overline H$
  • $H^-$

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.

$\operatorname{cl} \left({H}\right)$ and $\operatorname{Cl} \left({H}\right)$ are somewhat 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 $\operatorname{cl} \left({H}\right)$ can also be found in places.


Also see

  • Results about set closures can be found here.