Selective families of sets

Abstract Consider a cardinal number α, a set I and a family [Av:v in I) of sets. Suppose that for every subset N of I of cardinality less than α we are given a choice of an element x fNv A v for every v in N this paper the author investigates the circumstances under which it is then always possible to make a choice of an element x*of Avfor all v in which, in some precisely specified sense, can be approximated arbitrarily closely by some of the given partial choice functions x f . This question has turned out to be important when α is the least infinite cardinal number. Some of the results involve classes of ‘ large ’ cardinals.