cone-saturated

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In mathematics, specifically in order theory and functional analysis, if $C$ is a cone at 0 in a vector space $X$ such that $0\; \backslash in\; C,$ then a subset $S\; \backslash subseteq\; X$ is said to be $C$-saturated if $S\; =\; [S]\_C,$ where $[S]\_C\; :=\; (S\; +\; C)\; \backslash cap\; (S\; -\; C).$

Given a subset $S\; \backslash subseteq\; X,$ the $C$-saturated hull of $S$ is the smallest $C$-saturated subset of $X$ that contains $S.${{sfn|Schaefer|Wolff|1999|pp=215–222}}

If $\backslash mathcal\{F\}$ is a collection of subsets of $X$ then $\backslash left[\; \backslash mathcal\{F\}\; \backslash right]\_C\; :=\; \backslash left\backslash \{\; [F]\_C\; :\; F\; \backslash in\; \backslash mathcal\{F\}\; \backslash right\backslash \}.$

If $\backslash mathcal\{T\}$ is a collection of subsets of $X$ and if $\backslash mathcal\{F\}$ is a subset of $\backslash mathcal\{T\}$ then $\backslash mathcal\{F\}$ is a fundamental subfamily of $\backslash mathcal\{T\}$ if every $T\; \backslash in\; \backslash mathcal\{T\}$ is contained as a subset of some element of $\backslash mathcal\{F\}.$

If $\backslash mathcal\{G\}$ is a family of subsets of a TVS $X$ then a cone $C$ in $X$ is called a $\backslash mathcal\{G\}$-cone if $\backslash left\backslash \{\; \backslash overline\{[G]\_C\}\; :\; G\; \backslash in\; \backslash mathcal\{G\}\; \backslash right\backslash \}$ is a fundamental subfamily of $\backslash mathcal\{G\}$ and $C$ is a strict $\backslash mathcal\{G\}$-cone if $\backslash left\backslash \{\; [B]\_C\; :\; B\; \backslash in\; \backslash mathcal\{B\}\; \backslash right\backslash \}$ is a fundamental subfamily of $\backslash mathcal\{B\}.${{sfn|Schaefer|Wolff|1999|pp=215–222}}

$C$-saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices.