A correspondence is a set-valued function. That is, a correspondence associates to each point in one set a set of points in another set. As such, it can be viewed simply as a subset of the Cartesian product of the two sets. It may seem a bit silly to dedicate a chapter to such a topic, but correspondences arise naturally in many applications. For instance, the budget correspondence in economic theory associates the set of affordable consumption bundles to each price—income combination; the excess demand correspondence is a useful tool in studying economic equilibria; and the best-reply correspondence is the key to analyzing noncooperative games. The theory of “differential inclusions” deals with set-valued differential equations and plays an important role in control theory.
KeywordsTopological Space Compact Convex Topological Vector Space Polish Space Metrizable Space
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- 1.J. C. Moore  identifies five slightly different definitions of upper semicontinuity in use by economists, and points out some of the differences for compositions, etc.Google Scholar
- 2.The terms “multifunction” and “set-valued function” are used by some authors in place of “correspondence.”Google Scholar
- 3.If Y is Hausdorff, then the converse is true. That is, if φ is continuous at x 0, then each f i is continuous at x 0 . We leave the proof as an exercise.Google Scholar
- 4.N.B. Weak measurability has nothing to do with weak topologies.Google Scholar