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

Aapo 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 -