Star operation, microscopic sets and porous sets
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908672763887616 |
|---|---|
| author | Perkowska, Daria Żeberski, Szymon |
| author_facet | Perkowska, Daria Żeberski, Szymon |
| contents | 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 $\mathcal{P}(\mathbb{Z}), \mathcal{P}(\mathbb{Z}^ω),$ and $\mathcal{P}(2^ω)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19437 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Star operation, microscopic sets and porous sets Perkowska, Daria Żeberski, Szymon Logic 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 $\mathcal{P}(\mathbb{Z}), \mathcal{P}(\mathbb{Z}^ω),$ and $\mathcal{P}(2^ω)$ 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. |
| title | Star operation, microscopic sets and porous sets |
| topic | Logic |
| url | https://arxiv.org/abs/2510.19437 |