Classifying homogeneous cellular ordinal balleans up to coarse equivalence
Volume 149 / 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 $.
