Small uncountable cardinals in large-scale topology

DOI10.1016/J.TOPOL.2022.108277arXiv2002.08800MaRDI QIDQ6335228

Taras Banakh

Publication date: 20 February 2020

Abstract: In this paper we are interested in finding and evaluating cardinal characteristics of the continuum that appear in large-scale topology, usually as the smallest weights of coarse structures that belong to certain classes (indiscrete, inseparated, large) of finitary or locally finite coarse structures on $omega$. Besides well-known cardinals $mathfrakb,mathfrakd,mathfrakc$ we shall encounter two new cardinals $mathsfDelta$ and $mathsfSigma$, defined as the smallest weight of a finitary coarse structure on $omega$ which contains no discrete subspaces and no asymptotically separated sets, respectively. We prove that $maxmathfrakb,mathfraks,mathrmcov(mathcalN)lemathsfDeltalemathsfSigmalemathrmnon(mathcalM)$, but we do not know if the cardinals $mathsfDelta,mathsfSigma,mathrmnon(mathcalM)$ can be separated in suitable models of ZFC.

Mathematics Subject Classification ID

Combinatorics of partially ordered sets (06A07) Consistency and independence results (03E35) Cardinality properties (cardinal functions and inequalities, discrete subsets) (54A25) Consistency and independence results in general topology (54A35) Cardinal characteristics of the continuum (03E17) Ordered sets and their cofinalities; pcf theory (03E04)

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