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Register a new userSmall uncountable cardinals in large-scale topology
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Taras Banakh
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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 $mathfrak b,mathfrak d,mathfrak c$ we shall encounter two new cardinals $mathsf Delta$ and $mathsf Sigma$, 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 $max{mathfrak b,mathfrak s,mathrm{cov}(mathcal N)}lemathsf Deltalemathsf Sigmalemathrm{non}(mathcal M)$, but we do not know if the cardinals $mathsf Delta,mathsf Sigma,mathrm{non}(mathcal M)$ can be distinguished in suitable models of ZFC.
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