compactvalued

Compactvalued refers to a property of certain mathematical functions, particularly in the context of analysis and topology. A function is described as compactvalued if its image, for every point in its domain, is a compact set. Compactness is a fundamental concept in topology, generalizing the notion of closed and bounded intervals in Euclidean space to more abstract topological spaces.

For a function f: X -> Y, where X is a topological space and Y is a topological space, f is compactvalued if for every x in X, the set f(x) is compact in Y. This definition is most commonly encountered when the codomain Y is a Banach space or a more general topological vector space. In such spaces, a closed and bounded subset is not necessarily compact, making the condition of compactvaluedness a stronger requirement.

Compactvalued functions play a crucial role in the study of differential inclusions and control theory. They arise naturally when dealing with systems where the dynamics are not single-valued but rather a set of possible next states. The existence of solutions to such inclusions often relies on topological fixed-point theorems, such as the Michael selection theorem or theorems related to the properties of compactvalued maps. The interplay between the topology of the domain and the compactness of the image sets is key to establishing existence and properties of solutions.