Extremal balleans





Ballean, coarse structure, bornology, maximal ballean, ultranormal ballean, extremely normal ballean


A ballean (or coarse space) is a set endowed with a coarse structure. A ballean X is called normal if any two asymptotically disjoint subsets of X are asymptotically separated.  We say that a ballean X is ultra-normal (extremely normal) if any two unbounded subsets of X are not asymptotically disjoint (every unbounded subset of X is large).   Every maximal ballean is extremely normal and every extremely normal ballean is ultranormal, but the converse statements do not hold.   A normal ballean is ultranormal if and only if the Higson's corona of X is a singleton.   A discrete ballean X is ultranormal if and only if X is maximal.  We construct a series of concrete balleans with extremal properties.


Download data is not yet available.

Author Biography

Igor V. Protasov, Kyiv University

Faculty of Computer Science and Cybernetics



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

T. Banakh and I. Protasov, Constructing balleans, arXiv: 1812.03935.

D. Dikranjan, I. Protasov, K. Protasova and N. Zava, Balleans, hyperballeans and ideals, Appl. Gen. Topol, to appear.

M. Filali and I. Protasov, Slowly oscillating functions on locally compact groups, Appl.Gen. Topology 6 (2005), 67–77.

Ie. Lutsenko and I. Protasov, Thin subsets of balleans, Appl. Gen. Topol. 11 (2010), 89–93. https://doi.org/10.4995/agt.2010.1710

O. V. Petrenko and I. V. Protasov, Balleans and G-spaces, Ukr. Math. Zh. 64 (2012) ,344–350. https://doi.org/10.1007/s11253-012-0653-x

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

I. V. Protasov, Coronas of balleans, Topology Appl. 149 (2005), 149–160. https://doi.org/10.1016/j.topol.2004.09.005

I. V. Protasov, Asymptolic proximities, Appl. Gen. Topol. 9 (2008), 189–195. https://doi.org/10.4995/agt.2008.1799

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

I. Protasov and K. Protasova, Lattices of coarse structures, Math. Stud. 48 (2017),115–123. https://doi.org/10.15330/ms.48.2.115-123

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

O. Protasova, Maximal balleans, Appl. Gen. Topol. 7 (2006), 151–163. https://doi.org/10.4995/agt.2006.1920

J. Roe, Lectures on Coarse Geometry, AMS University Lecture Ser. 31, Providence, RI,2003.cAGT, https://doi.org/10.1090/ulect/031




How to Cite

I. V. Protasov, “Extremal balleans”, Appl. Gen. Topol., vol. 20, no. 1, pp. 297–305, Apr. 2019.