Note on the Banach Problem 1 of condensations of Banach spaces onto compacta

2021/12/20 by Osipov, Alexander V. · 1 citation
#FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2112.10718

Abstract

It is consistent with any possible value of the continuum \mathfrakc that every infinite-dimensional Banach space of density ≤ \mathfrakc condenses onto the Hilbert cube. Let μ be a cardinal of uncountable cofinality. It is consistent that the continuum be arbitrary large, no Banach space X of density γ, μ< γ< \mathfrakc condenses onto a compactum, but any Banach space of density μ admit a condensation onto a compactum. In particular, for μ=ω₁, it is consistent that \mathfrakc is arbitrarily large, no Banach space of density γ, ω₁< γ< \mathfrakc, condenses onto a compactum. These results imply a complete answer to the Problem 1 in the Scottish Book for Banach spaces: When does a Banach space X admit a bijective continuous mapping onto a compact metric space?

Cited by

• 90+ years of the Scottish Book

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