On hyperballeans of bounded geometry. (English) Zbl 1409.54010
A ball structure is a triple \(\mathcal{B}=(X,P,B)\), where \(X\), \(P\) are non-empty sets, and for all \(x\in X\) and \(\alpha \in P\), \(B(x,\alpha)\) is a subset of \(X\) which is called a ball of radius \(\alpha\) around \(x\). Let \(A\subseteq X\). Let \(B^*(x,\alpha)=\{y \in X: x\in B(y,\alpha)\}\), \(B(A,\alpha)=\bigcup_{a\in A} B(a,\alpha)\).
A ball structure \(\mathcal{B}=(X,P,B)\) is a ballean if
{Theorem 2.1.} For an unbounded ballean \(\mathcal{B}\) the following statements hold:
{Theorem 2.2}
Let \(\kappa \) be an infinite cardinal, \(n\in \mathbb{N}\), \([\kappa]^n=\{F\subset \kappa : |F|=n\}\), \(x\in \kappa\). Then the following statements hold:
A ball structure \(\mathcal{B}=(X,P,B)\) is a ballean if
- (i)
- for any \(\alpha, \beta, \in P\), there exists \(\alpha ', \beta ' \in P\) such that for every \(x\in X\), \(B(x,\alpha)\subseteq B^*(x,\alpha ')\), \(B^*(x,\beta)\subseteq B(x,\beta ')\);
- (ii)
- for every \(\alpha, \beta, \in P\), there exists \(\gamma \in P\) such that, for every \(x\in X\): \(B(B(x,\alpha), \beta)\subseteq B(x,\gamma)\);
- (iii)
- for any \(x,y\in X\) there exists \(\alpha \in P\), such that \(y\in B(x,\alpha)\).
{Theorem 2.1.} For an unbounded ballean \(\mathcal{B}\) the following statements hold:
- (i)
- \(\mathcal{B}^b\) is uniformly bounded locally finite if and only if \(\mathcal{B}=\mathbf{F}_X\);
- (ii)
- \(\mathcal{B}^b\) is of bounded geometry if and only if there exists a large subset \(Y\) of \(X\) such that \(\mathcal{B}_Y=\mathbf{F}_X\).
{Theorem 2.2}
Let \(\kappa \) be an infinite cardinal, \(n\in \mathbb{N}\), \([\kappa]^n=\{F\subset \kappa : |F|=n\}\), \(x\in \kappa\). Then the following statements hold:
- (i)
- The subballean of \(\mathbf{F}^b_{\kappa}\) with the support \([\kappa]^n\) is asymptotically scattered;
- (ii)
- The subballean of \(\mathbf{F}^b_{\kappa}\) with the support \(\{F \in \mathcal{F}_{\kappa}: x\in F\}\) is asymorphic to \(\mathbf{ Q}_{\kappa}\);
- (iii)
- \(\mathbf{F}^b_{\omega}\) can be partitioned into countably many pairwise close Cantor macrocubes but \(\mathbf{F}^b_{\omega}\) is not coarsely equivalent to \(\mathbf{Q}_{\omega}\).
Reviewer: Ana Pereira do Vale (Braga)
References:
[1] | Banakh, T.; Protasov, I.; Repovš, D.; Slobodianiuk, S., Classifying homogeneous cellular ordinal balleans up to coarse equivalence, Colloq. Math., 149, 211-224, (2017) · Zbl 1394.54015 · doi:10.4064/cm6785-4-2017 |
[2] | Banakh, TO; Protasov, IV; Slobodianiuk, SV, Scattered subsets of groups, Ukrainian Math. J., 67, 347-356, (2015) · Zbl 1359.20027 · doi:10.1007/s11253-015-1084-2 |
[3] | Banakh, T.; Zarichnyi, I., Characterizing the cantor bi-cube in asymptotic categories, Groups Geom. Dyn., 5, 691-728, (2011) · Zbl 1246.54023 · doi:10.4171/GGD/145 |
[4] | Dikranjan, D.; Zava, N., Some categorical aspects of coarse spaces and balleans, Topology Appl., 225, 164-194, (2017) · Zbl 1377.54027 · doi:10.1016/j.topol.2017.04.011 |
[5] | Dranishnikov, AN, Asymptotic topology, Russian Math. Surveys, 55, 1085-1129, (2000) · Zbl 1028.54032 · doi:10.1070/RM2000v055n06ABEH000334 |
[6] | Protasov, IV, Balleans of bounded geometry and \(G\)-spaces, Mat. Stud., 30, 61-66, (2008) · Zbl 1164.37301 |
[7] | Protasov, I., Banakh, T.: Ball Structures and Colorings of Graphs and Groups. Mathematical Studies Monograph Series, vol. 11. VNTL, L’viv (2003) · Zbl 1147.05033 |
[8] | Protasov, I., Zarichnyi, M.: General Asymptology. Mathematical Studies Monograph Series, vol. 12. VNTL, L’viv (2007) · Zbl 1172.54002 |
[9] | Roe, J.: Lectures on Coarse Geometry. University Lecture Series, vol. 31. American Mathematical Society, Providence (2003) · Zbl 1042.53027 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.