Year: 2020
Type: article
Abstract: We prove that every usco multimap $$\varPhi :X\rightarrow Y$$ Φ : X → Y from a metrizable separable space X to a GO-space Y has an $$F_\sigma $$ F σ -measurable selection. On the other hand, for the split interval $${\ddot{\mathbb I}}$$ I ¨ and the projection $$P:{{\ddot{\mathbb I}}}^2\rightarrow \mathbb I^2$$ P : I ¨ 2 → I 2 of its square onto the unit square $$\mathbb I^2$$ I 2 , the usco multimap $${P^{-1}:\mathbb I^2\multimap {{\ddot{\mathbb I}}}^2}$$ P - 1 : I 2 ⊸ I ¨ 2 has a Borel ( $$F_\sigma $$ F σ -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.
Source: Archive for Mathematical Logic
Author: Taras Banakh
Institutions: Jan Kochanowski University, Lviv University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,750
OpenAlex ID: W2948122914