On algebraic sums, trees and ideals in the Baire space
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914957973520384 |
|---|---|
| author | Mazurkiewicz, Łukasz Michalski, Marcin Rałowski, Robert Żeberski, Szymon |
| author_facet | Mazurkiewicz, Łukasz Michalski, Marcin Rałowski, Robert Żeberski, Szymon |
| contents | We work in the Baire space $\mathbb{Z}^ω$ equipped with the coordinate-wise addition $+$. Consider a $σ-$ideal $\mathcal{I}$ and a family $\mathbb{T}$ of some kind of perfect trees. We are interested in results of the form: for every $A\in \mathcal{I}$ and a tree $T\in\mathbb{T}$ there exists $T'\in \mathbb{T}, T'\subseteq T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\inω$. Explored tree types include perfect trees, uniformly perfect trees, Miller trees, Laver trees and $ω-$Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard $σ$-ideal $\mathcal{M}$ of meager sets, we also analyze $\mathcal{M}_-$ and fake null sets $\mathcal{N}$. The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On algebraic sums, trees and ideals in the Baire space Mazurkiewicz, Łukasz Michalski, Marcin Rałowski, Robert Żeberski, Szymon General Topology Primary: 03E75, 28A05, 54H05, Secondary: 03E17 We work in the Baire space $\mathbb{Z}^ω$ equipped with the coordinate-wise addition $+$. Consider a $σ-$ideal $\mathcal{I}$ and a family $\mathbb{T}$ of some kind of perfect trees. We are interested in results of the form: for every $A\in \mathcal{I}$ and a tree $T\in\mathbb{T}$ there exists $T'\in \mathbb{T}, T'\subseteq T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\inω$. Explored tree types include perfect trees, uniformly perfect trees, Miller trees, Laver trees and $ω-$Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard $σ$-ideal $\mathcal{M}$ of meager sets, we also analyze $\mathcal{M}_-$ and fake null sets $\mathcal{N}$. The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space. |
| title | On algebraic sums, trees and ideals in the Baire space |
| topic | General Topology Primary: 03E75, 28A05, 54H05, Secondary: 03E17 |
| url | https://arxiv.org/abs/2409.17748 |