ClosedselectedF

ClosedselectedF is a term used in the study of set-valued maps and selection problems. It denotes a family of selection operators that take a set-valued map S: X -> 2^Y and yield a single-valued function f: X -> Y, with emphasis on preserving closedness properties of the graph or the values. The "closed" in the name signals the focus on selections whose graphs are closed sets in the product topology, or on selections that arise as limits of other selections under a suitable topology.

Formally, let X be a topological space and Y a metric space. A set-valued map S: X -> 2^Y is nonempty-valued with closed values. A function f: X -> Y is a closed selection for S if f(x) in S(x) for all x and the graph Graph(f) = { (x,f(x)) ∈ X×Y : x ∈ X } is closed. The family closedselectedF(X,S) consists of all closed selections with additional closure properties, such as stability under taking pointwise limits of nets of selections that remain inside S.

Typical results describe conditions under which closed selections exist or can be approximated by continuous selections. Examples mirror classical selection theorems, including Michael's selection theorem, which provides continuous selections for certain lower semicontinuous set-valued maps with convex, closed values in Banach spaces. ClosedselectedF emphasizes the closed-graph aspect rather than continuity alone.

Example: X = R with standard topology, Y = R, S(x) = [x, x+1]. The function f(x) = x is a closed selection since f(x) ∈ S(x) and the graph is closed in R×R.

Applications appear in optimization, economics, and control theory where a deterministic choice is needed from a set of feasible options. Related concepts include set-valued maps, selection theorems, and the closed-graph property.