Borel's conjecture in topological groups

We introduce a natural generalization of Borel’s Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCא0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture,¬BCא1 is equivalent to the existence of a Kurepa tree of heightא1. Using the connection of BCκ with a generalization of Kurepa’s Hypothesis, we obtain the following consistency results: (1) If it is consistent that there is a 1-inaccessible cardinal then it is consistent that BCא1 . (2) If it is consistent that BCא1 , then it is consistent that there is an inaccessible cardinal. (3) If it is consistent that there is a 1-inaccessible cardinal with ω inaccessible cardinals above it, then ¬BCאω + (∀n < ω)BCאn is consistent. (4) If it is consistent that there is a 2-huge cardinal, then it is consistent that BCאω . (5) If it is consistent that there is a 3-huge cardinal, then it is consistent that BCκ for a proper class of cardinals κ of countable cofinality. A metric space (X, d) is strong measure zero if there is for each sequence ( n : n < ω) of positive real numbers a corresponding sequence (Un : n < ω) of open sets such that for each n the set Un has d-diameter at most n, and {Un : n < ω} covers X. Strong measure zero metric spaces are necessarily separable. E. Borel [5] conjectured that strong measure zero sets of real numbers are countable. The metric notion of strong measure zero has a natural generalization to non-metric contexts. Rothberger [17] introduced a generalization to the class of topological spaces. We consider a generalization to the class of topological groups. Most of our results can be presented in the more general context of uniformizable spaces, but we found no advantage to presenting it thus. Borel’s Conjecture also has natural generalizations to these non-metric contexts. These generalizations expose, as in the metric case, interesting connections with the foundations of mathematics. The generalization of Borel’s Conjecture considered here is quite different from what Halko and Shelah considered in [11]. After a brief introduction of notation and terminology we define Rothberger boundedness and Rothberger spaces in Section 1. In Section 2 we introduce a generalization of Borel’s Conjecture and in Section 3 explore connections between it and other combinatorial structures. In Section 4 we give a number of consistency and independence results regarding the generalization introduced in the earlier sections. By a well-known theorem of Kakutani a topological group is T0 if, and only if, it is T3 1 2 . Throughout this paper we shall assume, without further notice, that all groups considered are T3 1 2 . Correspondingly, all topological spaces we consider here are assumed to be T3 1 2 . Let (G, ∗) be a topological group with identity element 1. For nonempty subsets A and B of G and for g ∈ G the symbol A ∗ B denotes the set {a ∗ b : a ∈ A and b ∈ B}, and g ∗ A denotes {g} ∗ A. The symbol O denotes the set of all nonempty open covers of G. 2000Mathematics Subject Classification. Primary 03E05, Secondary 03E35, 03E55, 03E65, 22A99.