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Universal measure zero, large Hausdorff dimension, and nearly Lipschitz maps

Ondřej Zindulka (2012)

Fundamenta Mathematicae

We prove that each analytic set in ℝⁿ contains a universally null set of the same Hausdorff dimension and that each metric space contains a universally null set of Hausdorff dimension no less than the topological dimension of the space. Similar results also hold for universally meager sets. An essential part of the construction involves an analysis of Lipschitz-like mappings of separable metric spaces onto Cantor cubes and self-similar sets.

Universal spaces in the theory of transfinite dimension, I

Wojciech Olszewski (1994)

Fundamenta Mathematicae

R. Pol has shown that for every countable ordinal α, there exists a universal space for separable metrizable spaces X with ind X = α . We prove that for every countable limit ordinal λ, there is no universal space for separable metrizable spaces X with Ind X = λ. This implies that there is no universal space for compact metrizable spaces X with Ind X = λ. We also prove that there is no universal space for compact metrizable spaces X with ind X = λ.

Universal spaces in the theory of transfinite dimension, II

Wojciech Olszewski (1994)

Fundamenta Mathematicae

We construct a family of spaces with “nice” structure which is universal in the class of all compact metrizable spaces of large transfinite dimension ω 0 , or, equivalently, of small transfinite dimension ω 0 ; that is, the family consists of compact metrizable spaces whose transfinite dimension is ω 0 , and every compact metrizable space with transfinite dimension ω 0 is embeddable in a space of the family. We show that the least possible cardinality of such a universal family is equal to the least possible...

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