Classifying homogeneous cellular ordinal balleans up to coarse equivalence
Banakh, Taras, 1968- ...
For every ballean ▫$X$▫ we introduce two cardinal characteristics ▫$\text{cov}^\flat(X)$▫ and ▫$\text{cov}^\sharp(X)$▫ describing the capacity of balls in ▫$X$▫. We observe that these cardinal ...characteristics are invariant under coarse equivalence and prove that two cellular ordinal balleans ▫$X,Y$▫ are coarsely equivalent if ▫$\text{cof}(X)=\text{cof}(Y)$▫ and ▫$\text{cov}^\flat(X) = \text{cov}^\sharp(X) = \text{cov}^\flat(Y) = \text{cov}^\sharp(Y)$▫. This result implies that a cellular ordinal ballean ▫$X$▫ is homogeneous if and only if ▫$\text{cov}^\flat(X)=\text{cov}^\sharp(X)$▫. Moreover, two homogeneous cellular ordinal balleans ▫$X,Y$▫ are coarsely equivalent if and only if ▫$\text{cof}(X)=\text{cof}(Y)$▫ and ▫$\text{cov}^\sharp(X) = \text{cov}^\sharp(Y)$▫ if and only if each of these balleans coarsely embeds into the other ballean. This means that the coarse structure of a homogeneous cellular ordinal ballean ▫$X$▫ is fully determined by the values of the cardinals ▫$\text{cof}(X)▫$ and ▫$\text{cov}^\sharp(X)$▫. For every limit ordinal ▫$\gamma$▫ we shall define a ballean ▫$2^{<\gamma}$▫ (called the Cantor macro-cube), which in the class of cellular ordinal balleans of cofinality ▫$\text{cf}(\gamma)$▫ plays a role analogous to the role of the Cantor cube ▫$2^{\kappa}$▫ in the class of zero-dimensional compact Hausdorff spaces. We shall also present a characterization of balleans which are coarsely equivalent to ▫$2^{<\gamma}$▫. This characterization can be considered as an asymptotic analogue of Brouwer's characterization of the Cantor cube ▫$2^\omega$▫.
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