On coverings of Banach spaces and their subsets by hyperplanes

Abstract:

Given a Banach space we consider the $\sigma$-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the cofinality. We determine their values for separable Banach spaces, and approximate them for nonseparable Banach spaces. The remaining questions reduce to deciding if the following can be proved in ZFC (i.e. Zermelo-Fraenkel set theory with the axiom of choice) for every nonseparable Banach space $X$: (1) $X$ can be covered by $\omega _1$-many of its hyperplanes; (2) All subsets of $X$ of cardinalities less than $\operatorname {cf}([\operatorname {dens}(X)]^\omega )$ can be covered by countably many hyperplanes. We prove (1) and (2) for all Banach spaces in many well-investigated classes and that they are consistent with any possible size of the continuum. (1) is related to the problem whether every compact Hausdorff space which has small diagonal is metrizable and (2) to large cardinals.