Cone-saturated
# Cone-saturated

In mathematics, specifically in order theory and functional analysis, if $C$ is a cone at 0 in a vector space $X$ such that $0\in C,$ then a subset $S\subseteq X$ is said to be $C$-saturated if $S=[S]_{C},$ where $[S]_{C}:=(S+C)\cap (S-C).$ Given a subset $S\subseteq X,$ the $C$-saturated hull of $S$ is the smallest $C$-saturated subset of $X$ that contains $S.$ If ${\mathcal {F}}$ is a collection of subsets of $X$ then $\left[{\mathcal {F}}\right]_{C}:=\left\{[F]_{C}:F\in {\mathcal {F}}\right\}.$

If ${\mathcal {T}}$ is a collection of subsets of $X$ and if ${\mathcal {F}}$ is a subset of ${\mathcal {T}}$ then ${\mathcal {F}}$ is a fundamental subfamily of ${\mathcal {T}}$ if every $T\in {\mathcal {T}}$ is contained as a subset of some element of ${\mathcal {F}}.$ If ${\mathcal {G}}$ is a family of subsets of a TVS $X$ then a cone $C$ in $X$ is called a ${\mathcal {G}}$-cone if $\left\{{\overline {[G]_{C}}}:G\in {\mathcal {G}}\right\}$ is a fundamental subfamily of ${\mathcal {G}}$ and $C$ is a strict ${\mathcal {G}}$-cone if $\left\{[B]_{C}:B\in {\mathcal {B}}\right\}$ is a fundamental subfamily of ${\mathcal {B}}.$

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

## Properties

If $X$ is an ordered vector space with positive cone $C$ then $[S]_{C}=\bigcup \left\{[x,y]:x,y\in S\right\}.$

The map $S\mapsto [S]_{C}$ is increasing; that is, if $R\subseteq S$ then $[R]_{C}\subseteq [S]_{C}.$ If $S$ is convex then so is $[S]_{C}.$ When $X$ is considered as a vector field over $\mathbb {R} ,$ then if $S$ is balanced then so is $[S]_{C}.$

If ${\mathcal {F}}$ is a filter base (resp. a filter) in $X$ then the same is true of $\left[{\mathcal {F}}\right]_{C}:=\left\{[F]_{C}:F\in {\mathcal {F}}\right\}.$

