# Selection properties of the split interval and the Continuum Hypothesis

## TL;DR

This paper investigates the selection properties of upper semi-continuous multivalued maps, proving the existence of measurable selections in some cases and showing the dependence on the Continuum Hypothesis for others, highlighting limitations in extending known results.

## Contribution

It establishes the existence of $F_\sigma$-measurable selections for usco multimap into GO-spaces and demonstrates the CH-dependence of Borel selections for the split interval example.

## Key findings

- Every usco multimap from a metrizable separable space to a GO-space has an $F_\sigma$-measurable selection.
- The existence of Borel selections for a specific usco multimap on the split interval depends on the Continuum Hypothesis.
- Results on Borel selections into fragmentable compact spaces cannot be generalized to all compact spaces.

## Abstract

We prove that every usco multimap $\Phi:X\to Y$ 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 $P:\ddot{\mathbb I}^2\to{\mathbb I}^2$ of its square onto the unit square ${\mathbb I}^2$, the usco multimap $P^{-1}:{\mathbb I}^2\multimap\ddot{\mathbb I}^2$ 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.