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Classifying homogeneous cellular ordinal balleans up to coarse equivalence
ID
Banakh, Taras
(
Avtor
),
ID
Protasov, Igorʹ Vladimirovič
(
Avtor
),
ID
Repovš, Dušan
(
Avtor
),
ID
Slobodianiuk, Sergiy
(
Avtor
)
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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 ▫$2^\omega$▫.
Jezik:
Angleški jezik
Ključne besede:
coarse space
,
ballean
,
cellular ballean
,
ordinal ballean
,
homogeneous ballean
,
coarse equivalence
,
cellular entourage
,
asymptotic dimension
,
Cantor macro-cube
Vrsta gradiva:
Članek v reviji
Tipologija:
1.01 - Izvirni znanstveni članek
Organizacija:
PEF - Pedagoška fakulteta
FMF - Fakulteta za matematiko in fiziko
Leto izida:
2017
Št. strani:
Str. 211-224
Številčenje:
Vol. 149, no. 2
PID:
20.500.12556/RUL-109506
UDK:
515.124
ISSN pri članku:
0010-1354
DOI:
10.4064/cm6785-4-2017
COBISS.SI-ID:
18045529
Datum objave v RUL:
04.09.2019
Število ogledov:
925
Število prenosov:
447
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Objavi na:
Gradivo je del revije
Naslov:
Colloquium mathematicum
Skrajšan naslov:
Colloq. math.
Založnik:
Éditions Scientifiques de Pologne
ISSN:
0010-1354
COBISS.SI-ID:
25242624
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