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

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Halko A., Shelah S.

Fundamenta Mathematicae

2001

tom 170

nr 3

219-229

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].

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Halko A.

1 Shelah S.

1 2001

Wyniki wyszukiwania

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