INSERTIQ_NOF A CONTINUOUS FUNCTION AND X x I

If g and f are real-valued functions defined on a topological space X such that g < f (i.e., g(x) < f(x) for each x in X), consider the problem of finding necessary and sufficient conditions in order for there to be a continuous function h defined on X such that g < h < f. Such condi tions seem to have characteristics that keep them from being applicable in some cases. For example, the condition given in Theorem 3.11 of [2] requires that-g must possess the same property as f and that the classes to which g and f belong are closed under sum, supremum, and infimum. The condition given in Theorem 2.1 of [5] places a restriction on f- g. This paper gives a necessary and sufficient con dition that avoids these problems, but it is in terms of subsets of X x I. All functions considered are assumed to map into the interval (0,1); I denotes the interval [0,1]. (In most situations there is no loss in generality in replacing R with (0,1).) It is therefore convenient to let C(X) represent the set of all continuous functions from X into (0,1). The characterization that is given below is in terms of the following subsets of X x I: If k maps X into (0,1), let lThis work was supported by a released teaching loa