A NOTE ON SEMI-CONTINUOUS SET-VALUED FUNCTIONS

Abstract

Let X be metric, Y compact metric and I(Y) the space of all non-empty closed subsets of Y. Equip I(Y) with the Vietoris topology. The following result can be found in [3], p. 70: Each semi-continuous function F: X → I(Y) is of Baire class 1 (i.e. inverse images of open sets are F_o-sets). In [4], p. 47, Kuratowski posed the following question: Can metrizability and compactness of the spaces X and Y respectively be replaced by weaker assumptions? The aim of this note is to answer the question affirmatively.