On strong measure zero subsets of ^k2

We study the generalized Cantor space ^K2 and the generalized Baire space ^KK as analogues of the classical Cantor and Baire spaces. We equip ^KK with the topology where a basic neighborhood of a point η is the set {v : (∀j < i)(v(j) = η(j))}, where i < K. We define the concept of a strong measure zero set of ^K2. We prove for successor K = K^<K that the ideal of strong measure zero sets of ^K2 is b_K-additive, where b_K is the size of the smallest unbounded family in ^KK, and that the generalized Borel conjecture for ^K2 is false. Moreover, for regular uncountable K, the family of subsets of ^K2 with the property of Baire is not closed under the Suslin operation. These results answer problems posed in [2].