Borel measurable Hahn-Mazurkiewicz theorem

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 me...

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Bibliographic Details
Published inTopology and its applications Vol. 333; p. 108536
Main Authors Dudák, Jan, Vejnar, Benjamin
Format Journal Article
LanguageEnglish
Published Elsevier B.V 15.06.2023
Subjects
Arc
Borel measurability
Cantor space
Group action
Hyperspace
Peano continuum
secondary
Hyperspace
Borel measurability
Group action
Arc
Peano continuum
Cantor space
primary
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Summary: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.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2023.108536