# On strong measure zero subsets of ${}^{\kappa}2$

Fundamenta Mathematicae (2001)

- Volume: 170, Issue: 3, page 219-229
- ISSN: 0016-2736

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topAapo Halko, and Saharon Shelah. "On strong measure zero subsets of $^{κ}2$." Fundamenta Mathematicae 170.3 (2001): 219-229. <http://eudml.org/doc/281783>.

@article{AapoHalko2001,

abstract = {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].},

author = {Aapo Halko, Saharon Shelah},

journal = {Fundamenta Mathematicae},

keywords = {strong measure zero; -additive; uncountable regular cardinals; bounding number; dominating number; generalized Borel conjecture},

language = {eng},

number = {3},

pages = {219-229},

title = {On strong measure zero subsets of $^\{κ\}2$},

url = {http://eudml.org/doc/281783},

volume = {170},

year = {2001},

}

TY - JOUR

AU - Aapo Halko

AU - Saharon Shelah

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

JO - Fundamenta Mathematicae

PY - 2001

VL - 170

IS - 3

SP - 219

EP - 229

AB - 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].

LA - eng

KW - strong measure zero; -additive; uncountable regular cardinals; bounding number; dominating number; generalized Borel conjecture

UR - http://eudml.org/doc/281783

ER -

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