The γ-Borel conjecture

In this paper we prove that it is consistent that every γ-set is countable while not every strong measure zero set is countable. We also show that it is consistent that every strong γ-set is countable while not every γ-set is countable. On the other hand we show that every strong measure zero set is countable iff every set with the Rothberger property is countable. A set of realsX has strong measure zero iff for any sequence ( n : n < ω) of positive reals there exists a sequence of intervals (In : n < ω) covering X with each In of length less than n. Laver [8] showed that it is relatively consistent with ZFC that the Borel conjecture is true, i.e., every strong measure zero set is countable. Sets of reals called γ-sets were first considered by Gerlits and Nagy [5]. They showed that every γ set has strong measure zero and that Martin’s Axiom implies every set of reals of size smaller than the continuum is a γset. A γ-set of size continuum is constructed in Galvin and Miller [4] using MA. Next we define γ-set. An open cover U of a topological space X is an ω-cover iff for every finite F ⊆ X there exists U ∈ U with F ⊆ U and X / ∈ U . An open cover U of X is a γ-cover iff U is infinite and each x ∈ X is in all but finitely many U ∈ U . Finally, X is a γ-set iff X is a separable metric space in which every ω-cover contains a γ-subcover. Paul Szeptycki asked if it was possible to have a sort of weak Borel conjecture be true, i.e., every γ-set countable, while the Borel conjecture is false. We answer his question positively. We use Hechler [6] forcing, H, for adding a dominating real, an analysis of it due to Baumgartner and Dordal [1], properties of Laver forcing L, and a characterization of H due to Truss [11]. Thanks to Boise State University for support during the time this paper was written and to Alan Dow for some helpful discussions and to Boaz Tsaban for some suggestions to improve an earlier version. Mathematics Subject Classification 2000: 03E35; 03E17