Orbitally discrete coarse spaces

Igor V. Protasov

doi:10.4995/agt.2021.13874

Authors

Igor V. Protasov Taras Shevchenko National University of Kyiv https://orcid.org/0000-0003-1518-6234

DOI:

https://doi.org/10.4995/agt.2021.13874

Keywords:

coarse space, ultrafilter, orbitally discrete space, almost finitary space, scattered space

Abstract

Given a coarse space (X, E), we endow X with the discrete topology and denote X â™¯ = {p âˆˆ βG : each member P âˆˆ p is unbounded }. For p, q âˆˆ X â™¯ , p||q means that there exists an entourage E âˆˆ E such that E[P] âˆˆ q for each P âˆˆ p. We say that (X, E) is orbitally discrete if, for every p âˆˆ X â™¯ , the orbit p = {q âˆˆ X â™¯ : p||q} is discrete in βG. We prove that every orbitally discrete space is almost finitary and scattered.

Downloads

Download data is not yet available.

Author Biography

Igor V. Protasov, Taras Shevchenko National University of Kyiv

Professor, Faculty of Computer Science and Cybernetics

References

T. Banakh and I. Protasov, Set-theoretical problems in Asymptology,arXiv: 2004.01979.

T. Banakh, I. Protasov and S. Slobodianiuk, Scattered subsets of groups, Ukr. Math. J. 67 (2015), 347-356. https://doi.org/10.1007/s11253-015-1084-2

T. Banakh and I. Zarichnyi, Characterizing the Cantor bi-cube in asymptotic categories, Groups Geom. Dyn. 5, no. 4 (2011), 691-728. https://doi.org/10.4171/GGD/145

Ie. Lutsenko and I. Protasov, Space, thin and other subsets of groups, Intern. J. Algebra Comput. 19 (2009), 491-510. https://doi.org/10.1142/S0218196709005135

Ie. Lutsenko and I. V. Protasov, Thin subset of balleans, Appl. Gen. Topology 11 (2010), 89-93. https://doi.org/10.4995/agt.2010.1710

O. Petrenko and I. V. Protasov, Balleans and filters, Mat. Stud. 38 (2012), 3-11. https://doi.org/10.1007/s11253-012-0653-x

I. V. Protasov, Normal ball structures, Mat. Stud. 20 (2003), 3-16.

I. V. Protasov, Balleans of bounded geometry and G-spaces, Algebra Dicrete Math. 7, no. 2 (2008), 101-108.

I. V. Protasov, Sparse and thin metric spaces, Mat. Stud. 41 (2014), 92-100.

I. Protasov, Decompositions of set-valued mappings, Algebra Discrete Math. 30, no. 2 (2020), 235-238. https://doi.org/10.12958/adm1485

I. Protasov, Coarse spaces, ultrafilters and dynamical systems, Topol. Proc. 57 (2021), 137-148.

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

I. Protasov and K. Protasova, Lattices of coarse structures, Math. Stud. 48 (2017), 115-123.

I. V. Protasov and S. Slobodianiuk, Thin subsets of groups, Ukrain. Math. J. 65 (2013), 1245-1253. https://doi.org/10.1007/s11253-014-0866-2

I. Protasov and S. Slobodianiuk, On the subset combinatorics of $G$-spaces, Algebra Dicrete Math. 17, no. 1 (2014), 98-109.

I. Protasov and S. Slobodianiuk, Ultracompanions of subsets of a group, Comment. Math. Univ. Carolin. 55, no. 1 (2014), 257-265.

I. Protasov and S. Slobodianiuk, The dynamical look at the subsets of groups, Appl. Gen. Topology 16 (2015), 217-224. https://doi.org/10.4995/agt.2015.3584

I. Protasov and M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., Vol. 12, VNTL, Lviv, 2007.

J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, American Mathematical Society, Providence RI, 2003. https://doi.org/10.1090/ulect/031