Extremal balleans

Authors

DOI:

https://doi.org/10.4995/agt.2019.11260

Keywords:

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

Abstract

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.

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Author Biography

Igor V. Protasov, Kyiv University

Faculty of Computer Science and Cybernetics

Professor

References

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Published

2019-04-01

How to Cite

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

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Articles