SetValued

Setvalued, often written set-valued, refers to a mapping that assigns to each input a set of outputs. Formally, if X and Y are sets, a setvalued map F assigns x ↦ F(x) with F(x) ⊆ Y for every x ∈ X. This is typically denoted F: X → P(Y), where P(Y) is the power set of Y. When F(x) ≠ ∅ for all x, F is nonempty-valued; dom(F) = { x ∈ X : F(x) ≠ ∅ }. The graph is Graph(F) = { (x, y) : y ∈ F(x) } ⊆ X × Y. If every F(x) is a singleton, F reduces to an ordinary function.

Topology and analysis extend setvalued maps to X and Y as topological or metric spaces. Regularity notions include upper semicontinuity and lower semicontinuity; closed-valued or convex-valued variants aid analysis. Operations are defined pointwise, for example (F ∪ G)(x) = F(x) ∪ G(x) and (F ∩ G)(x) = F(x) ∩ G(x).

A selection of F is a single-valued function f: X → Y with f(x) ∈ F(x) for all x. Existence of selections, especially continuous or measurable ones, is a central topic; results such as Michael's selection theorem give conditions under which selections exist.

Applications appear in optimization with set constraints, differential inclusions dx/dt ∈ F(x), control theory, economics, and viability theory. Example: F(x) = [0, x] for x ≥ 0 and F(x) = [x, 0] for x < 0. In short, setvalued maps generalize ordinary functions by allowing multiple outputs per input.