Definition:Closure (Topology)/Definition 2
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
The closure of $H$ (in $T$) is defined as:
- $\ds H^- := \bigcap \leftset {K \supseteq H: K}$ is closed in $\rightset T$
That is, $H^-$ is the intersection of all closed sets in $T$ which contain $H$.
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
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.5$: Topological spaces