English

Selection properties of the split interval and the Continuum Hypothesis

General Topology 2021-11-01 v1 Logic

Abstract

We prove that every usco multimap Φ:XY\Phi:X\to Y from a metrizable separable space XX to a GO-space YY has an FσF_\sigma-measurable selection. On the other hand, for the split interval I¨\ddot{\mathbb I} and the projection P:I¨2I2P:\ddot{\mathbb I}^2\to{\mathbb I}^2 of its square onto the unit square I2{\mathbb I}^2, the usco multimap P1:I2I¨2P^{-1}:{\mathbb I}^2\multimap\ddot{\mathbb I}^2 has a Borel (FσF_\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.

Keywords

Cite

@article{arxiv.1905.02243,
  title  = {Selection properties of the split interval and the Continuum Hypothesis},
  author = {Taras Banakh},
  journal= {arXiv preprint arXiv:1905.02243},
  year   = {2021}
}

Comments

10 pages

R2 v1 2026-06-23T08:58:33.863Z