×

Selectors and orderings of coarse spaces. (English. Ukrainian original) Zbl 1471.54009

J. Math. Sci., New York 256, No. 6, 779-784 (2021); translation from Ukr. Mat. Visn. 18, No. 1, 71-79 (2021).
In this paper, the relationship between selectors and linear orders on coarse spaces is clarified. A coarse structure on a set \(X\) is a filter \(\mathcal{E}\) on \(X \times X\) that includes the diagonal set and is closed under composition and inverse. The pair \((X, \mathcal{E})\) is called a coarse space. A map \(f\colon X\to Y\) between coarse spaces is said to be macro-uniform (or bornologous) if \((f\times f)(E) \in \mathcal{E}_{Y}\) for all \(E \in \mathcal{E}_X\). These notions, introduced independently by [I. Protasov and T. Banakh, Ball Structures and Colorings of Graphs and Groups. Lviv: VNTL Publishers (2003; Zbl 1147.05033)] and [J. Roe, Lectures on Coarse Geometry. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1042.53027)], are asymptotic counterparts of uniform spaces and uniformly continuous maps. Given a coarse space \((X, \mathcal{E})\), the family of non-empty subsets of \(X\) can be considered as a coarse space \((\exp X, \exp \mathcal{E})\) in a canonical way. For a non-empty subspace \(\mathcal{F}\) of \(\exp X\), a macro-uniform choice function of \(\mathcal{F}\) is called an \(\mathcal{F}\)-selector. An \(\mathcal{F}\)-selector is also called a 2-selector, a bornologous selector and a global selector if \(\mathcal{F} = \left[X\right]^{2}\) (the family of unordered pairs), \(\mathcal{F} = \mathcal{B}_{X}\) (the family of bounded subsets) and \(\mathcal{F} = \exp X\), respectively. It has been shown in [I. Protasov, “Selectors of discrete coarse spaces”, Comment. Math. Univ. Carolin (to appear; arXiv:2101.07199)] that a discrete coarse space admits a 2-selector if and only if there exists a linear order \(\leq\) on \(X\) such that \(\mathcal{B}_{X}\) coincides with the family of bounded subsets with respect to \(\leq\).
In Section 2, the author analyses the interrelation between selectors and linear orders. The main results of this section include the following:
Let \(X\) be coarse space with a compatible linear order \(\leq\). The binary minimum function \(\min\colon \left[X\right]^{2} \to X\) is a 2-selector. (Proposition 2)
Let \(X\) be coarse space with a compatible well-order \(\leq\). The minimum function \(\min\colon \exp X \to X\) is a global selector. (Proposition 3)
It is also proved that the existence of a global (2- and bornologous) selector is invariant under coarse equivalence (Proposition 5).
In Section 3, some existence and non-existence results on global selectors of cellular coarse spaces are provided. Recall that a coarse structure is said to be cellular if it has a base consisting of equivalence relations, and is said to be ordinal if it has a base well-ordered by inclusion. By Proposition 3, every cellular ordinal space admits a global selector (Theorem 3). Let \(G\) be a group endowed with the coarse structure generated by the sets of the form \(\left\{(x, y) \in G\times G \colon y \in Fx \right\}\), \(e_{G} \in F \subseteq_{\mathrm{fin}} X\). If \(G\) is uncountable, then it does not admit a 2-selector (Theorem 4). On the other hand, if \(G\) is countable and locally finite, then it admits a global selector (Theorem 5).
In Section 4, the author concludes the paper with the discussion on two coarse structures of \(\omega\). A binary relation \(E\) on a set \(X\) is said to be locally finite if \(E[x]\) and \(E^{-1}[x]\) are finite for each \(x \in X\), and is said to be finitary if \(\sup_{x\in X} \left|E[x]\right|, \sup_{x \in X} \left|E^{-1}[x]\right| < \infty\). Then the family \(\Lambda\) (resp. \(\mathcal{F}\)) of locally finite (resp. finitary) binary relations on \(\omega\) forms a coarse structure. The space \((\omega, \Lambda)\) admits a global selector (Theorem 6), while the space \((\omega, \mathcal{F})\) does not admit a 2-selector (Theorem 7).

MSC:

54C65 Selections in general topology
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Banakh, T.; Protasov, I.; Repovs, D.; Slobodianiuk, S., Classifying homogeneous cellular ordinal balleans up to coarse equivalence, Colloquium Mathematicum, 149, 211-224 (2017) · Zbl 1394.54015 · doi:10.4064/cm6785-4-2017
[2] Banakh, T.; Zarichnyi, I., Characterizing the Cantor bi-cube in asymptotic categories, Groups Geom. Dynam., 5, 691-728 (2011) · Zbl 1246.54023 · doi:10.4171/GGD/145
[3] Y. Cornulier and P. de la Harpe, “Metric geometry of locally compact groups,” Colloquium Mathematic in Mathematics,25, Zürich (2016). · Zbl 1352.22001
[4] Dikranjan, D.; Protasov, I.; Protasova, K.; Zava, N., Balleans, hyperballeans and ideals, Appl. Gen. Topology, 2, 431-447 (2019) · Zbl 1429.54035 · doi:10.4995/agt.2019.11645
[5] P. de la Harpe, Topics in Geometrical Group Theory. University Chicago Press (2000). · Zbl 0965.20025
[6] Protasov, IV, Morphisms of ball’s structures of groups and graphs, Ukrain. Mat. Zh., 54, 847-855 (2008) · Zbl 1003.05053
[7] Protasov, IV, Balleans of bounded geometry and G-spaces, Algebra Discrete Math., 7, 2, 101-108 (2008) · Zbl 1164.37302
[8] I. Protasov, “Selectors of discrete coarse spaces,” Comment.Math. Univ. Carolin. preprint arXiv: 2101.07199 (to appear). · Zbl 1432.54031
[9] I. Protasov and T. Banakh, Ball Structures and Colorings of Groups and Graphs. Math. Stud. Monogr. Ser., vol. 11. VNTL, Lviv (2003). · Zbl 1147.05033
[10] I. Protasov and K. Protasova, “On hyperballeans of bounded geometry,” Europ. J. Math., 4, 1515-1520 (2018). · Zbl 1409.54010
[11] I. Protasov and K. Protasova, “The normality of macrocubes and hiperballeans,” Europ. J. Math. (2020). doi:10.1007/s40879-020-00400-x.
[12] I. Protasov and M. Zarichnyi, General Asymptology. Math. Stud. Monogr. Ser., vol. 12. VNTL, Lviv (2007). · Zbl 1172.54002
[13] J. Roe, Lectures on Coarse Geometry. Univ. Lecture Ser., vol. 31. American Mathematical Society, Providence RI (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.