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

Aapo Halko ; Saharon Shelah

Fundamenta Mathematicae, Tome 167 (2001), p. 219-229 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $_{κ}$-additive, where ${}_{κ}$ 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].

Détail

Publié le : 2001-01-01

Zbl 0994.03038

EUDML-ID : urn:eudml:doc:281783