On strong measure zero subsets of $^{κ}2$

• Autorzy: Aapo Halko , Saharon Shelah
• Języki publikacji: EN

Abstrakty

EN

We study the generalized Cantor space $^{κ}2$ and the generalized Baire space $^{κ}κ$ as analogues of the classical Cantor and Baire spaces. We equip $^{κ}κ$ with the topology where a basic neighborhood of a point η is the set {ν: (∀j < i)(ν(j) = η(j))}, where i < κ.
We define the concept of a strong measure zero set of $^{κ}2$. We prove for successor $κ = κ^{<κ}$ that the ideal of strong measure zero sets of $^{κ}2$ is $𝔟_{κ}$-additive, where ${𝔟}_{κ}$ is the size of the smallest unbounded family in $^{κ}κ$, and that the generalized Borel conjecture for $^{κ}2$ is false. Moreover, for regular uncountable κ, the family of subsets of $^{κ}2$ with the property of Baire is not closed under the Suslin operation.
These results answer problems posed in [2].

Kategorie tematyczne

• 03E05: Other combinatorial set theory
• 03E15: Descriptive set theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2001
• Tom: 170
• Numer: 3
• Strony: 219-229

Daty

wydano

2001

Twórcy

autor

Aapo Halko

• Department of Mathematics, P.O. Box 4, FIN-00014 University of Helsinki, Helsinki, Finland

autor

Saharon Shelah

• Institute of Mathematics, Hebrew University, Jerusalem, Israel

Identyfikatory

DOI

10.4064/fm170-3-1