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ジタさん (selbstdenker@mathtod.online)'s status on Sunday, 22-Oct-2023 17:45:56 JST ジタさん

《命題：無限次元可分ヒルベルト空間は $2^{\aleph_0}$ 個未満の超平面で覆えない》最初は楽勝でZFCから証明できると思っていたが、自信がなくなってきた。CHやMAを仮定すれば一発なのだが、CPAあたりから否定命題が証明できたりするんだろうか。

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  fujidig@mathtod.online's status on Sunday, 22-Oct-2023 17:45:56 JST fujidig
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  @selbstdenker この論文に答えが載っていました。 https://arxiv.org/abs/2103.05097
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    On coverings of Banach spaces and their subsets by hyperplanes
    
    Given a Banach space we consider the $σ$-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 for every nonseparable Banach space $X$: (1) $X$ can be covered by $ω_1$-many of its hyperplanes; (2) All subsets of $X$ of cardinalities less than ${\rm cf}([{\rm dens}(X)]^ω)$ 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.
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  fujidig@mathtod.online's status on Sunday, 22-Oct-2023 18:08:10 JST fujidig
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  @selbstdenker ジタさんの設定では必ずしも原点を通らない超平面を考えていて、この論文では必ず原点を通る超平面を考えているという差異があるかもしれません。
  
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  ジタさん (selbstdenker@mathtod.online)'s status on Sunday, 22-Oct-2023 21:27:31 JST ジタさん
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  @fujidig 情報ありがとうございます。なんでこんな話をしているかについては京都で時間があればお話しします。
  
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