Star operation, microscopic sets and porous sets

TL;DR

This paper investigates the properties and relationships of star operations, microscopic sets, and porous sets, providing new constructions and theoretical insights into their interactions and implications under the Borel Conjecture.

Contribution

It introduces new results on the star operation $ ext{F}^*$, constructs specific families satisfying $ ext{F} = ext{F}^*$, and analyzes the interplay between microscopic and porous sets.

Findings

Characterization of $ ext{F}^{**} = ext{F}$ under certain conditions

Construction of families satisfying $ ext{F} = ext{F}^*$ in various power sets

Insights into the relationship between microscopic and porous sets

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.