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In mathematics, specifically in order theory and functional analysis, if C {\displaystyle C} is a cone at 0 in a vector space X {\displaystyle X} such that 0 ∈ C , {\displaystyle 0\in C,} then a subset S ⊆ X {\displaystyle S\subseteq X} is said to be C {\displaystyle C} -saturated if S = C , {\displaystyle S=_{C},} where C := ∩ . {\displaystyle _{C}:=\cap.} Given a subset S ⊆ X , {\displaystyle S\subseteq X,} the C {\displaystyle C} -saturated hull of S {\displaystyle S} is the smallest C {\displaystyle C} -saturated subset of X {\displaystyle X} that contains S . {\displaystyle S.} If F {\displaystyle {\mathcal {F}}} is a collection of subsets of X {\displaystyle X} then C := { C : F ∈ F } . {\displaystyle \left_{C}:=\left\{_{C}:F\in {\mathcal {F}}\right\}.}
If T {\displaystyle {\mathcal {T}}} is a collection of subsets of X {\displaystyle X} and if F {\displaystyle {\mathcal {F}}} is a subset of T {\displaystyle {\mathcal {T}}} then F {\displaystyle {\mathcal {F}}} is a fundamental subfamily of T {\displaystyle {\mathcal {T}}} if every T ∈ T {\displaystyle T\in {\mathcal {T}}} is contained as a subset of some element of F . {\displaystyle {\mathcal {F}}.} If G {\displaystyle {\mathcal {G}}} is a family of subsets of a TVS X {\displaystyle X} then a cone C {\displaystyle C} in X {\displaystyle X} is called a G {\displaystyle {\mathcal {G}}} -cone if { C ¯ : G ∈ G } {\displaystyle \left\{{\overline {_{C}}}:G\in {\mathcal {G}}\right\}} is a fundamental subfamily of G {\displaystyle {\mathcal {G}}} and C {\displaystyle C} is a strict G {\displaystyle {\mathcal {G}}} -cone if { C : B ∈ B } {\displaystyle \left\{_{C}:B\in {\mathcal {B}}\right\}} is a fundamental subfamily of B . {\displaystyle {\mathcal {B}}.}
C {\displaystyle C} -saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices.