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Classifying homogeneous cellular ordinal balleans up to coarse equivalence

Volume 149 / 2017

T. Banakh, I. Protasov, D. Repovš, S. Slobodianiuk Colloquium Mathematicum 149 (2017), 211-224 MSC: Primary 54E35; Secondary 51F99. DOI: 10.4064/cm6785-4-2017 Published online: 29 May 2017

Abstract

For every ballean $X$, we introduce two cardinal characteristics $\mathrm {cov}^\flat (X)$ and $\mathrm {cov}^\sharp (X)$ describing the capacity of balls in $X$. We observe that these characteristics are invariant under coarse equivalence and prove that two cellular ordinal balleans $X,Y$ are coarsely equivalent if $\mathrm {cof}(X)=\mathrm {cof}(Y)$ and $\mathrm {cov}^\flat (X)=\mathrm {cov}^\sharp (X)=\mathrm {cov}^\flat (Y)=\mathrm {cov}^\sharp (Y)$. This implies that a cellular ordinal ballean $X$ is homogeneous if and only if $\mathrm {cov}^\flat (X)=\mathrm {cov}^\sharp (X)$. Moreover, two homogeneous cellular ordinal balleans $X,Y$ are coarsely equivalent if and only if $\mathrm {cof}(X)=\mathrm {cof}(Y)$ and $\mathrm {cov}^\sharp (X)=\mathrm {cov}^\sharp (Y)$ if and only if each of these balleans coarsely embeds into the other. This means that the coarse structure of a homogeneous cellular ordinal ballean $X$ is fully determined by the values of $\mathrm {cof}(X)$ and $\mathrm {cov}^\sharp (X)$. For every limit ordinal $\gamma $, we define a ballean $2^{ \lt \gamma }$ (called the Cantor macro-cube) that, in the class of cellular ordinal balleans of cofinality $\mathrm {cf}(\gamma )$, plays a role analogous to the role of the Cantor cube $2^{\kappa }$ in the class of zero-dimensional compact Hausdorff spaces. We also characterize balleans which are coarsely equivalent to $2^{ \lt \gamma }$. This can be considered as an asymptotic analogue of Brouwer’s characterization of the Cantor cube $2^\omega $.

Authors

  • T. BanakhFaculty of Mechanics and Mathematics
    Ivan Franko National University of Lviv
    Lviv, Ukraine
    and
    Institute of Mathematics
    Jan Kochanowski University
    Kielce, Poland
    e-mail
  • I. ProtasovFaculty of Cybernetics
    Taras Shevchenko National University of Kyiv
    Kyiv, Ukraine
    e-mail
  • D. RepovšFaculty of Education
    and Faculty of Mathematics and Physics
    University of Ljubljana
    Kardeljeva Pl. 16
    Ljubljana, Slovenia 1000
    e-mail
  • S. SlobodianiukFaculty of Cybernetics
    Taras Shevchenko National University of Kyiv
    Kyiv, Ukraine
    e-mail

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