On strong measure zero subsets of
Volume 170 / 2001
Abstract
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] .
