# Definition:Closure Operator/Notation

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## Definition

Let $T = \struct {S, \preccurlyeq}$ or $T = \struct {S, \tau}$ be an ordered structure or topological space.

Let $H \subseteq S$.

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.

## Sources

- 2013: Francis Clarke:
*Functional Analysis, Calculus of Variations and Optimal Control*... (previous) ... (next): $1.1$: Basic Definitions