Balleans, hyperballeans and ideals

Dikran Dikranjan, Igor V. Protasov, Ksenia Protasova, Nicolò Zava

Abstract

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.


Keywords

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

Subject classification

54E15

Full Text:

PDF

References

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. https://doi.org/10.4171/GGD/145

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. https://doi.org/10.1016/j.topol.2017.04.011

D. Dikranjan and N. Zava, Preservation and reflection of size properties of balleans, Topology Appl. 221 (2017), 570--595. https://doi.org/10.1016/j.topol.2017.02.008

A. Dow, Closures of discrete sets in compact spaces, Studia Sci. Math. Hungar. 42, no. 2 (2005), 227--234. https://doi.org/10.1556/SScMath.42.2005.2.7

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. https://doi.org/10.1007/s11253-012-0653-x

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. https://doi.org/10.1090/ulect/031

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

Abstract Views

1491
Metrics Loading ...

Metrics powered by PLOS ALM


 

Cited-By (articles included in Crossref)

This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site

1. The normality of macrocubes and hyperballeans
Igor Protasov, Ksenia Protasova
European Journal of Mathematics  vol: 7  issue: 3  first page: 1274  year: 2021  
doi: 10.1007/s40879-020-00440-x



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt