Keywords: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.
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