Dense Continuity and Selections of Set-Valued Mappings

Abstract

A theorem proved by Fort in 1951 says that an upper or lower semi-continuous set-valued mapping from a Baire space A into non-empty compact subsets of a metric space is both lower and upper semi-continuous at the points of a dense Gδ -subset of A. In this paper we show that the conclusion of Fort’s theorem holds under the weaker hypothesis of either upper or lower quasi-continuity. The existence of densely defined continuous selections for lower quasi-continuous mappings is also proved.