Topological diagonalizations and Hausdorff dimension

doi:10.1285/i15900932v22n2p83

Topological diagonalizations and Hausdorff dimension

Tomasz Weiss, Boaz Tsaban

Abstract

The Hausdorff dimension of a product $X\times Y$ can be strictly greater than that of $Y$ , even when the Hausdorff dimension of $X$ is zero. But when $X$ is countable, the Hausdorff dimensions of $Y$ and $X\times Y$ are the same. Diagonalizations of covers define a natural hierarchy of properties which are weaker than "being countable" and stronger than "having Hausdorff dimension zero". Fremlin asked whether it is enough for $X$ to have the strongest property in this hierarchy (namely, being a $\gamma$ -set) in order to assure that the Hausdorff dimensions of $Y$ and $X\times Y$ are the same.
We give a negative answer: Assuming the Continuum Hypothesis, there exists a $\gamma$ -set $X \subseteq \mathbb{R}$ and a set $Y \subseteq \mathbb{R}$ with Hausdorff dimension zero, such that the Hausdorff dimension of $X+Y$ (a Lipschitz image of $X\times Y$ ) is maximal, that is, $1$ . However, we show that for the notion of a strong $\gamma$ -set the answer is positive. Some related problems remain open.

DOI Code: 10.1285/i15900932v22n2p83

Keywords: Hausdorff dimension; Gerlits-Nagy $gamma$ property; Galvin-Miller strong $gamma$ property

Classification: 03E75; 37F20; 26A03

Full Text: PDF

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This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.

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Topological diagonalizations and Hausdorff dimension

Abstract