Closeness and linkness in balleans

  • I.V. Protasov Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
  • K. Protasova Faculty of Computer Science and Cybernetics Kyiv University
Keywords: Coarse structure; ballean; coarse space; closeness; linkness; rigidity; stability


A set $X$ endowed with a coarse structure is called ballean or coarse space. For a ballean $(X, \mathcal{E})$, we say that two subsets $A$, $B$ of $X$ are close (linked) if there exists an entourage $E\in \mathcal{E}$ such that $A\subseteq E [B]$, $B\subseteq E[A]$ (either $A, B$ are bounded or contain unbounded close subsets). We explore the following general question: which information about a ballean is contained and can be extracted from the relations of closeness and linkness.


T. Banakh, Small uncountable cardinals in large-scale topology, preprint

T. Banakh, I. Protasov, Functional boundedness of balleans: coarse versions of compactness, Axioms, 2019, 8, 33;

T. Banakh, I. Protasov, The normality and bounded growth of balleans, arXiv: 1810.07979.

T. Banakh, I. Protasov, Minmax bornologies, Ukr. Math. Bull., 16 (2019), 496–502.

D. Dikranjan, I. Protasov, K. Protasova, N. Zava, Balleans, hyperballeans and ideals, Appl. Gen. Topology, 20 (2019), 431–447.

O. Petrenko, I. Protasov, Balleans and filters, Math. Stud., 38 (2012), 3–11.

I. Protasov, Normal ball structures, Math. Stud., 20 (2003), 3–16.

I. Protasov, Coronas of balleans, Topology Applications, 149 (2005), 149–160.

I. Protasov, Asymptotic proximities, Appl. Gen. Topology, 9 (2008), 189–195.

I.V. Protasov, Balleans of bounded geometry and G-space, Algebra Discrete Math., 2008, №2, 101–108.

I. Protasov, T. Banakh, Ball structures and colorings of groups and graphs, Math. Stud. Monogr. Ser., V.11, VNTL, Lviv, 2003.

I. Protasov, K. Protasova, On hyperballeans of bounded geometry, Europ. J. Math., 4 (2018), 1515–1520.

I. Protasov, M. Zarichnyi, General Asymptopogy, Math. Stud. Monogr., V.12, VNTL, Lviv, 2007.

J. Roe, Lectures on Coarse Geometry, AMS University Lecture Ser., V.31, Providence, RI, 2003.

How to Cite
Protasov I, Protasova K. Closeness and linkness in balleans. Mat. Stud. [Internet]. 2020Mar.17 [cited 2021Oct.16];53(1):100-8. Available from: