Qenclosed

Qenclosed is a term used in set theory and topology to describe a general way of capturing the idea that a subset is contained within a chosen family of enclosing sets. Let X be a set and Q be a nonempty family of subsets of X, viewed as admissible enclosures. The Q-enclosure of a subset S ⊆ X is defined as the intersection of all enclosures in Q that contain S: cl_Q(S) = ⋂{E ∈ Q : S ⊆ E}. A subset S is called Q-enclosed if S = cl_Q(S).

If X is itself an enclosure in Q (i.e., X ∈ Q) and Q is closed under finite intersections, then cl_Q behaves as a closure operator: it is extensive, monotone, and idempotent. Consequently, the collection of Q-enclosed subsets of X forms a closure system, also known as a Moore family, with respect to this operator.

Examples help illustrate the concept. If Q is the family of all closed sets in a given topology on X, then cl_Q(S) coincides with the usual topological closure of S, and Q-enclosed sets are precisely the closed sets. If X is a real vector space and Q is the family of all convex subsets of X, then cl_Q(S) is the convex hull of S, the smallest convex set containing S.

Qenclosed thus provides a unifying framework for various hull-like constructions, including closures and hulls, under a single operator defined by a chosen enclosure family. It connects to theories of closure operators, lattices, and convexity spaces.

See also: closure operator, closed set, convex hull, hull, Moore family, envelope.