Continuous selections and realcompactness

Abstract

The class of basically fixed, lowersemicontinuous carriersis defined, and the existence of continuous selections formembers of this class is investigated. It is shown that,barring the existence of measurable cardinals, a completelyregular Hausdorff space is realcompact iff every basicallyfixed, lowersemicontinuous carrier of infinite character fromthe space to the convex subsets of a locally convex spaceadmits a selection. One application of this result is theproof that the union of a locally finite collection of real-compact cozero sets is realcompact, provided the union isof nonmeasurable cardinal.