Undetermined sets of point-open games

We show that a set of reals is undetermined in Galvin’s point-open game iff it is uncountable and has property C′′, which answers a question of Gruenhage. Let X be a topological space. The point-open game G(X) of Galvin [G] is played as follows. Black chooses a point x0 ∈ X, then White chooses an open set U0 3 x0, then B chooses a point x1 ∈ X, then W chooses an open set U1 3 x1, etc. B wins the play (x0, U0, x1, U1, . . .) iff X = ⋃ n Un. Galvin [G] showed that the Continuum Hypothesis yields a Lusin set X which is undetermined (i.e. for which the game G(X) is undetermined). (A Lusin set is an uncountable set of reals which has countable intersection with every meager set.) Recently Rec law [R] showed that every Lusin set is undetermined. Motivated by Rec law’s result we prove the following. Theorem. Let X be a topological space in which every point is Gδ. Then G(X) is undetermined iff X is uncountable and has property C ′′. Property C ′′ was introduced by Rothberger (see [M]). A topological space X has property C ′′ if for every sequence Un (n ∈ ω) of open covers of X there exist Un ∈ Un such that X = ⋃ n Un. It is known (see [M] or [FM]) that every Lusin set has property C ′′. Clearly, a space with property C ′′ must be Lindelöf. Martin’s Axiom implies that every Lindelöf space of size less than 2א0 has property C ′′ and that there are sets of reals of size 2א0 with property C ′′ (see [M]). Thus, Martin’s Axiom yields undetermined sets of reals of size 2א0 (Theorem 4 of [G]). On the other hand, in Laver’s [L] model for Borel’s conjecture all metric spaces with property C ′′ are countable (see Note 1). Thus, consistently, all metric spaces are determined. 1991 Mathematics Subject Classification: 03E15, 54G15. Supported by KBN grant PB 2 1017 91 01.