Classifying homogeneous cellular ordinal balleans up to coarse equivalence

Banakh, Taras (author), Protasov, Igorʹ Vladimirovič (author), Repovš, Dušan (author), Slobodianiuk, Sergiy (author)

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 ▫$2^\omega$▫.

• Language: English
• Keywords: coarse space, ballean, cellular ballean, ordinal ballean, homogeneous ballean, coarse equivalence, cellular entourage, asymptotic dimension, Cantor macro-cube
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2017
• Number of pages: Str. 211-224
• Numbering: Vol. 149, no. 2
• PID: 20.500.12556/RUL-109506
• UDC: 515.124
• ISSN on article: 0010-1354
• DOI: 10.4064/cm6785-4-2017
• COBISS.SI-ID: 18045529
• Publication date in RUL: 04.09.2019

Record is a part of a journal

• Title: Colloquium mathematicum
• Shortened title: Colloq. math.
• Publisher: Éditions Scientifiques de Pologne
• ISSN: 0010-1354
• COBISS.SI-ID: 25242624