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

Aapo Halko ¹   ; Saharon Shelah ²

¹ Department of Mathematics P.O. Box 4 FIN-00014 University of Helsinki Helsinki, Finland
² Institute of Mathematics Hebrew University Jerusalem, Israel

Fundamenta Mathematicae, Tome 170 (2001) no. 3, pp. 219-229

Résumé

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

Détail

DOI : 10.4064/fm170-3-1

Keywords: study generalized cantor space kappa generalized baire space kappa kappa analogues classical cantor baire spaces equip kappa kappa topology where basic neighborhood point eta set forall eta where kappa define concept strong measure zero set kappa prove successor kappa kappa kappa ideal strong measure zero sets kappa frak kappa additive where frak kappa size smallest unbounded family kappa kappa generalized borel conjecture kappa false moreover regular uncountable kappa family subsets kappa property baire closed under suslin operation these results answer problems posed