A multifunction generalization of Gale's Ascoli theorem

Abstract

Our purpose is to improve the Gale-type multifunction Ascoli theorem of Mancuso (1971; page 470). This latter supposes the range space to be normal and Hausdorff, and therefore does not contain Gale's theorem (1950; page 304). To obtain a multifunction theorem containing Gale's theorem (also Mancuso's theorem), we return to Gale's essential hypotheses. Thus, we assume the regularity of the range space in the sufficiency direction, and, in the necessity direction, weassume the domain to be a k-space and the range to be a regular Hausdorff space. We dispense with the “point-like” condition imposed by Mancuso. Unexplained terminology and notation is that of Mancusopos;s paper.