Borel Measurable Hahn-Mazurkiewicz Theorem

TL;DR

This paper demonstrates that the classical Hahn-Mazurkiewicz theorem can be realized through Borel measurable assignments, providing measurable selections of continuous mappings onto Peano continua and other compact metric spaces.

Contribution

It introduces Borel measurable constructions for continuous mappings onto Peano continua and compact metric spaces, extending classical topological results with measure-theoretic methods.

Findings

Borel measurable assignment of continuous maps onto Peano continua

Borel measurable assignment of continuous maps from Cantor set onto compact spaces

Measurable arc assignment in Peano continua

Abstract

It is well known due to Hahn and Mazurkiewicz that every Peano continuum is a continuous image of the unit interval. We prove that an assignment, which takes as an input a Peano continuum and produces as an output a continuous mapping whose range is the Peano continuum, can be realized in a Borel measurable way. Similarly, we find a Borel measurable assignment which takes any nonempty compact metric space and assigns a continuous mapping from the Cantor set onto that space. To this end we use the Burgess selection theorem. Finally, a Borel measurable way of assigning an arc joining two selected points in a Peano continuum is found.