Balleans, hyperballeans and ideals


  • Dikran Dikranjan Università di Udine
  • Igor V. Protasov Kyiv University
  • Ksenia Protasova Kyiv University
  • Nicolò Zava Udine University



balleans, coarse structure, coarse map, asymorphism, balleans defined by ideals, hyperballeans


A ballean B (or a coarse structure) on a set X is a family of subsets of X called balls (or entourages of the diagonal in X × X) dened in such a way that B can be considered as the asymptotic counterpart of a uniform topological space. The aim of this paper is to study two concrete balleans dened by the ideals in the Boolean algebra of all subsets of X and their hyperballeans, with particular emphasis on their connectedness structure, more specically the number of their connected components.


Download data is not yet available.

Author Biographies

Dikran Dikranjan, Università di Udine

Department of Mathematics and Computer Science

Igor V. Protasov, Kyiv University

Department of Computer Science and Cybernetics

Ksenia Protasova, Kyiv University

Department of Computer Science and Cybernetics

Nicolò Zava, Udine University

Department of Mathematics and Computer Science


T. Banakh, I. Protasov, D. Repovs and S. Slobodianiuk, Classifying homogeneous cellular ordinal balleans up to coarse equivalence, arxiv: 1409.3910v2.

T. Banakh and I. Zarichnyi, Characterizing the Cantor bi-cube in asymptotic categories, Groups, Geometry and Dynamics 5 (2011), 691-728.

W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Grundlehren der mathematischen Wissenschaften, Band 211, Springer--Verlag, Berlin-Heidelberg-New York, 1974.

D. Dikranjan and N. Zava, Some categorical aspects of coarse structures and balleans, Topology Appl. 225 (2017), 164--194.

D. Dikranjan and N. Zava, Preservation and reflection of size properties of balleans, Topology Appl. 221 (2017), 570--595.

A. Dow, Closures of discrete sets in compact spaces, Studia Sci. Math. Hungar. 42, no. 2 (2005), 227--234.

K. Kunen, Set theory. An introduction to independence proofs, Studies in Logic and Foundations of Math., vol. 102, North-Holland, Amsterdam-New York-Oxford, 1980.

O. Petrenko and I. Protasov, Balleans and filters, Mat. Stud. 38, no. 1 (2012), 3--11.

I. Protasov and T. Banakh, Ball Structures and Colorings of Groups and Graphs, Mat. Stud. Monogr. Ser 11, VNTL, Lviv, 2003.

I. Protasov and K. Protasova, On hyperballeans of bounded geometry, arXiv:1702.07941v1.

I. Protasov and M. Zarichnyi, General Asymptology, 2007 VNTL Publishers, Lviv, Ukraine.

J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, American Mathematical Society, Providence RI, 2003.

N. Zava, On F-hyperballeans, work in progress.




How to Cite

D. Dikranjan, I. V. Protasov, K. Protasova, and N. Zava, “Balleans, hyperballeans and ideals”, Appl. Gen. Topol., vol. 20, no. 2, pp. 431–447, Oct. 2019.



Regular Articles