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Covering the real line with translates of a zero-dimensional compact set

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Máthé A.

Fundamenta Mathematicae

2011

tom 213

nr 3

213-219

We construct a compact set C of Hausdorff dimension zero such that cof(𝒩) many translates of C cover the real line. Hence it is consistent with ZFC that less than continuum many translates of a zero-dimensional compact set can cover the real line. This answers a question of Dan Mauldin.

Can we assign the Borel hulls in a monotone way?

63%

Elekes M., Máthé A.

Fundamenta Mathematicae

2009

tom 205

nr 2

105-115

A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/$G_{δ}$ hull to every negligible/measurable subset of [0,1]? Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone $G_{δ}$ hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent. We also answer the question of Z. Gyenes and D. Pálvölgyi whether monotone hulls can be defined for every chain of measurable sets. Moreover, we comment on the problem of hulls of all subsets of [0,1].

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2 Fundamenta Mathematicae

2 Máthé A.

1 Elekes M.

1 2011

1 2009

Wyniki wyszukiwania

Covering the real line with translates of a zero-dimensional compact set

Can we assign the Borel hulls in a monotone way?