Topologies on Closed and Closed Convex Sets pp 183-234 | Cite as

# Multifunctions: the Rudiments

## Abstract

If *f* is a function from a set *Y* to a set *X*,then for each *x* ∈ *X* there may be several points of *Y* sent by *f* to *x*. Thus the inverse assignment *x* → {y ∈ *Y*: *f*(*y*) = *x*} is naturally multi-valued, and unless *f* is onto, some of the values of the inverse are empty. Only when *f* is a bijection will this assignment be a single-valued function in the usual sense. In this chapter we present the elements of the theory of functions that assign to points of a given set *X* subsets of a second set Y. Such functions are called set-valued functions or multifunctions. There are a number of set-valued functions that naturally arise in Banach space geometry, some of which we introduce here.

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