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Recovering a compact Hausdorff space $X$ from the compatibility ordering on $C(X)$

Volume 242 / 2018

Tomasz Kania, Martin Rmoutil Fundamenta Mathematicae 242 (2018), 187-205 MSC: Primary 46E10, 46T20; Secondary 06F25. DOI: 10.4064/fm388-11-2017 Published online: 11 May 2018

Abstract

Let $f$ and $g$ be scalar-valued, continuous functions on some topological space. We say that $g$ dominates $f$ in the compatibility ordering if $g$ coincides with $f$ on the support of $f$. We prove that two compact Hausdorff spaces are homeomorphic if and only if there exists a compatibility isomorphism between their families of scalar-valued, continuous functions. We derive the classical theorems of Gelfand–Kolmogorov, Milgram and Kaplansky as easy corollaries to our result, as well as a theorem of Jarosz [Bull. Canad. Math. Soc. 33 (1990)]. Sharp automatic-continuity results for compatibility isomorphisms are also established.

Authors

  • Tomasz KaniaMathematics Institute
    University of Warwick
    Gibbet Hill Rd
    Coventry, CV4 7AL, England
    and
    Institute of Mathematics
    Czech Academy of Sciences
    Žitná 25
    115 67 Praha 1, Czech Republic
    e-mail
  • Martin RmoutilDepartment of Mathematics Education
    Faculty of Mathematics and Physics
    Charles University
    Sokolovská 83
    186 75 Praha 8, Czech Republic
    e-mail

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