Extremal balleans

Igor V. Protasov


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.


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

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Universitat Politècnica de València

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