Cardinal estimates involving the weak Lindelöf game

We show that if X is a first-countable Urysohn space where player II has a winning strategy in the game G(1)(omega 1) (O, O-D) (the weak Lindelof game of length omega(1)) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelof game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a winning strategy in the game G(fin)(omega 1) (O, O-D), providing some partial answers to it. We finish by constructing an example of a compact space where player II does not have a winning strategy in the weak Lindelof game of length omega(1).