Selection properties of the split interval and the Continuum Hypothesis

Abstract

We prove that every usco multimap varPhi :Xrightarrow Y from a metrizable separable space X to a GO-space Y has an F_sigma -measurable selection. On the other hand, for the split interval {ddot{mathbb I}} and the projection P:{{ddot{mathbb I}}}^2rightarrow mathbb I^2 of its square onto the unit square mathbb I^2, the usco multimap {P^{-1}:mathbb I^2multimap {{ddot{mathbb I}}}^2} has a Borel (F_sigma -measurable) selection if and only if the Continuum Hypothesis holds. This CH-example shows that know results on Borel selections of usco maps into fragmentable compact spaces cannot be extended to a wider class of compact spaces.