Classifying homogeneous cellular ordinal balleans up to coarse equivalence

Taras Banakh (Avtor), Igor´ Vladimirovič Protasov (Avtor), Dušan Repovš (Avtor), Sergiy Slobodianiuk (Avtor)

Povzetek

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 ▫$2^\omega$▫.

Ključne besede

coarse space;ballean;cellular ballean;ordinal ballean;homogeneous ballean;coarse equivalence;cellular entourage;asymptotic dimension;Cantor macro-cube;

Podatki

Jezik:	Angleški jezik
Leto izida:	2017
Tipologija:	1.01 - Izvirni znanstveni članek
Organizacija:	UL FMF - Fakulteta za matematiko in fiziko
UDK:	515.124
COBISS:	18045529
ISSN:	0010-1354
Št. ogledov:	462
Št. prenosov:	291
Ocena:	0 (0 glasov)
Metapodatki:

Ostali podatki

Vrsta dela (COBISS):	Članek v reviji
Strani:	str. 211-224
Letnik:	ǂVol. ǂ149
Zvezek:	ǂno. ǂ2
Čas izdaje:	2017
DOI:	10.4064/cm6785-4-2017
ID:	11215357