In mathematics, specifically in order theory and functional analysis, if C is a cone at 0 in a vector space X such that 0 ∈ C, then a subset S of X is said to be C-saturated if S = [S]C, where [S]C := (S + C) ∩ (S − C). Given a subset S of X, the C-saturated hull of S is the smallest C-saturated subset of X that contains S. If is a collection of subsets of X in X then . If is a collection of subsets of X and if is a subset of then is a fundamental subfamily of if every is contained as a subset of some element of . If is a family of subsets of a TVS X then a cone C in X is called a -cone if is a fundamental subfamily of and C is a strict -cone if is a fundamental subfamily of . C-saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices.