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arXiv · 2510.19437

Star operation, microscopic sets and porous sets

Daria Perkowska·Szymon Żeberski
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Abstract

This paper explores the interplay between star operations, microscopic sets, and porous sets. The study focuses on the Galvin-Mycielski-Solovay theorem, which characterizes strongly measure zero sets and their interactions with meager sets. Results include the investigation of the star operation $\mathcal{F}^*$ and its properties. The paper also examines the relationship between porous sets and microscopic sets. Additionally, the work presents constructions of families $\mathcal{F}$ in $\mathcal{P}(\mathbb{Z}), \mathcal{P}(\mathbb{Z}^\omega),$ and $\mathcal{P}(2^\omega)$ that satisfy $\mathcal{F} = \mathcal{F}^*$. Theorems and lemmas are provided to establish conditions under which $\mathcal{F}^{**} = \mathcal{F}$ and to analyze the implications of the Borel Conjecture and its dual. The paper concludes with a discussion of microscopic sets and their properties, including their interactions with porous sets and the non-equivalence of certain classes of sets.

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Daria Perkowska, Szymon Żeberski. 2025-10-22. Star operation, microscopic sets and porous sets. https://arxiv.org/abs/2510.19437

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