Classifying homogeneous cellular ordinal balleans up to coarse equivalence

T. Banakh¹ ; I. Protasov² ; D. Repovš³ ; S. Slobodianiuk²

¹ Faculty of Mechanics and Mathematics Ivan Franko National University of Lviv Lviv, Ukraine and Institute of Mathematics Jan Kochanowski University Kielce, Poland
² Faculty of Cybernetics Taras Shevchenko National University of Kyiv Kyiv, Ukraine
³ Faculty of Education and Faculty of Mathematics and Physics University of Ljubljana Kardeljeva Pl. 16 Ljubljana, Slovenia 1000

Colloquium Mathematicum, Tome 149 (2017) no. 2, pp. 211-224.

Résumé

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 $.

Détail

DOI : 10.4064/cm6785-4-2017

Keywords: every ballean introduce cardinal characteristics mathrm cov flat mathrm cov sharp describing capacity balls observe these characteristics invariant under coarse equivalence prove cellular ordinal balleans coarsely equivalent mathrm cof mathrm cof mathrm cov flat mathrm cov sharp mathrm cov flat mathrm cov sharp implies cellular ordinal ballean homogeneous only mathrm cov flat mathrm cov sharp moreover homogeneous cellular ordinal balleans coarsely equivalent only mathrm cof mathrm cof mathrm cov sharp mathrm cov sharp only each these balleans coarsely embeds other means coarse structure homogeneous cellular ordinal ballean fully determined values mathrm cof mathrm cov sharp every limit ordinal gamma define ballean gamma called cantor macro cube class cellular ordinal balleans cofinality mathrm gamma plays role analogous role cantor cube kappa class zero dimensional compact hausdorff spaces characterize balleans which coarsely equivalent gamma considered asymptotic analogue brouwer characterization cantor cube omega