On Strong Measure Zero Subsets of ^{κ}2

Aapo Halko

This joint work with Saharon Shelah.

We define the natural concept of the strong measure zero in the generalized Cantor space ^{κ}2 and the generalized Baire space ^{κ}κ. 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. In this paper we answer some problems posed in [1].

Theorem A The family of strong measure zero subsets of ^{κ}2 is 2^{κ}-additive under GMA(κ) and 2^{<κ} = κ.

Theorem B The generalized Borel conjecture is false in ^{κ}2 assuming ZFC+κ^{<κ} = κ.

[1] A. Halko. Negligible subsets of the generalized Baire space ω_{1}^{ω_{1}}, Ann. Acad. Sci. Ser. Diss. Math. 108 (1996).

Address: Department of Mathematics, P.O. BOX 4, FIN-00014 University of Helsinki, Helsinki, Finland

E-mail: aapo.halko@helsinki.fi