Continuous selections and realcompactness

Ilya Eugene Blum and Srinivasa Swaminathan

Vol. 93 (1981), No. 2, 251–260

DOI: 10.2140/pjm.1981.93.251

Abstract

The class of basically fixed, lowersemicontinuous carriers is defined, and the existence of continuous selections for members of this class is investigated. It is shown that, barring the existence of measurable cardinals, a completely regular Hausdorff space is realcompact iff every basically fixed, lowersemicontinuous carrier of infinite character from the space to the convex subsets of a locally convex space admits a selection. One application of this result is the proof that the union of a locally finite collection of realcompact cozero sets is realcompact, provided the union is of nonmeasurable cardinal.

Mathematical Subject Classification 2000

Primary: 54C65

Secondary: 54D60

Milestones

Received: 10 October 1979

Revised: 16 May 1980

Published: 1 April 1981