A local to global selection theorem for simplex-valued functions

Suppose we are given a function : X ! K where X is a paracompact space and K is a simplicial complex, and an open cover fU j 2 g of X, so that for each 2 , f : U ! jKj is a map that is a selection of on its domain. We shall prove that there is a map f : X ! jKj which is a selection of . We shall also show that under certain conditions on such a set of maps or on the complex K, there exists a : X ! K with the property that each f is a selection of on its domain and that there is a selection f : X ! jKj of . The term selection, as used herein, will always refer to a map f, i.e., continuous function, having the property that f(x) 2 (x) for each x in the domain.