arXiv:math/9710218v1 [math.LO] 15 Oct 1997 On strong measure zero subsets of κ 2 Aapo Halko ∗ Saharon Shelah † September 25, 2013 ‡ Abstract This paper answers three questions posed in [1]. Theorem 2.7 The family of strong measure zero subsets of ω 1 2 is 2 ℵ 1 -additive under GMA and CH. Theorem 3.1 The generalized Borel conjecture is false in ω 1 2 assum- ing ZFC+CH. Theorem 4.2 The family of subsets of ω 1 2 with the property of Baire is not closed under the Souslin operation. 1 Introduction We study the generalized Cantor space κ 2 and the generalized Baire space κ κ for an uncountable cardinal 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 ∈ κ κ :(∀j<i)(ν (j )= η(j ))}, where i<κ. A systematic study of measure and category in these spaces was started in [1]. In this paper we answer some problems posed in [1]. * Partially supported by grant #1011049 of the Academy of Finland † Publication number 662 ‡ 1991 Mathematics Subject Classification: 03E15, 04A15. 1