2019/01/13 by Ігор Протасов, Igor Protasov, Protasov, Igor
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Mathematical Dynamics and Fractals #math.GN
paper · pdf · doi:10.48550/arxiv.1901.03977
Ballean, coarse structure, bornology, maximal ballean, ultranormal ballean, extremely normal ballean
openalex publication_date 2019/01/13 · arxiv created 2019/01/21 · arxiv updated 2019/01/23 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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 ultranormal (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.