\mathop\rm PL₊(I) is not a Polish group

2014/07/30 by Michael P. Cohen, Cohen, Michael P., Robert R. Kallman +1
Decision Sciences · Mathematics · #54H05 #54H11 #57S05 #Advanced Topology and Set Theory #FOS: Mathematics #Fuzzy and Soft Set Theory #Group Theory (math.GR) #Mathematical Dynamics and Fractals #math.GR #msc:54H05 #msc:54H11 #msc:57S05

paper · pdf · doi:10.48550/arxiv.1407.7910

19 pages, 2 figures

openalex publication_date 2014/07/30 · arxiv created 2014/08/07 · arxiv updated 2014/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The group \mathop\rm PL₊(I) of increasing piecewise linear self-homeomorphisms of the interval I=[0,1] may not be assigned a topology in such a way that it becomes a Polish group. The same statement holds for the groups \mathop\rm Homeo₊^Lip(I) of bi-Lipschitz homeomorphisms of I, and \mathop\rm Diff₊^1+ε(I) of diffeomorphisms of I whose derivatives are Hölder continuous with exponent ε.

Related

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• Classification and Non-classification of Homeomorphism Relations
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