closedvalued

Closed-valued is a term used in set-valued analysis to describe a multifunction whose values are closed sets in the codomain space. If F is a set-valued map from a space X to subsets of a space Y, F is called closed-valued when for every x in X, the set F(x) is a closed subset of Y.

In many applications, F is assumed to be nonempty-valued, meaning F(x) ≠ ∅ for all x. However, closed-valued does not by itself guarantee nonemptiness. The concept is most often discussed in topological or metric spaces, where the notion of closeness and convergence supports limit arguments used in analysis.

Examples include a constant-valued map F(x) = C for all x, where C is a fixed closed subset of Y, which is trivially closed-valued. More generally, any map defined by a closed predicate or via a closed construction on the input can yield closed-valued outputs.

Closed-valued multifunctions are common in optimization, variational analysis, differential inclusions, viability theory, and equilibrium problems, where constraints or feasible sets are modeled as F(x). The closedness of the valued sets often ensures stability under limits and aids in proving existence results or convergence of iterative methods.

Relationship to continuity properties is important: closed-valuedness concerns the nature of the values, while notions like upper semicontinuity and lower semicontinuity concern how those values change with x. Together, these properties help characterize the graph and behavior of the multifunction F.

See also: set-valued map, multivalued analysis, upper semicontinuity, closed set.