Extremal balleans

• Main Author: Igor Protasov
• Format: Article
• Language: English
• Published: Universitat Politècnica de València 2019-04-01
• Series: Applied General Topology
• Subjects: Ballean; coarse structure; bornology; maximal ballean; ultranormal ballean; extremely normal ballean
• Online Access: https://polipapers.upv.es/index.php/AGT/article/view/11260

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.