There are no hereditary productive γ-spaces

We show that if X is an uncountable productive γ-set F. Jordan, Productive local properties of function spaces, Topology Appl. 154 (2007) 870-883], then there is a countable Y⊆X such that X?Y is not Hurewicz.Along the way we answer a question of A. Miller by showing that an increasing countable union of γ-spaces is again a γ-space. We will also show that λ-spaces with the Hurewicz property are precisely those spaces for which every co-countable set is Hurewicz.