Selective covering properties of product spaces, II: γ spaces

We study productive properties of γ spaces and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: (1) Solving a problem of F. Jordan, we show that for every unbounded tower set X ⊆ R of cardinality ℵ1, the space C_p(X) is productively Fréchet– Urysohn. In particular, the set X is productively γ. (2) Solving problems of Scheepers and Weiss and proving a conjecture of Babinkostova–Scheepers, we prove that, assuming the Continuum Hypothesis, there are γ spaces whose product is not even Menger. (3) Solving a problem of Scheepers–Tall, we show that the properties γ and Gerlits–Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)). We apply our results to solve a large number of additional problems and use Arhangel’skiĭ duality to obtain results concerning local properties of function spaces and countable topological groups.