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arXiv · 2506.22592

Separating covering numbers and separating uniformity numbers of Hausdorff measures need large continuum

Tatsuya Goto
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Abstract

We show that if $\mathfrak{c} = \aleph_2$ then all covering numbers of Hausdorff measures $\operatorname{cov}(\mathcal{N}^s(\mathbb{R}^d))$ ($0 < s < d, d \in \omega$) are equal and all uniformity numbers $\operatorname{non}(\mathcal{N}^s(\mathbb{R}^d))$ ($0 < s < d, d \in \omega$) are equal. This is a partial answer to Problem 5.3 and 5.4 of [4].

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Tatsuya Goto. 2025-06-27. Separating covering numbers and separating uniformity numbers of Hausdorff measures need large continuum. https://arxiv.org/abs/2506.22592

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