Abstract: | By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space X is covered by a Kσ subset of X if and only if it does not contain a closed-in-X subset homeomorphic to the Baire space ww. We consider the analogous statement (which we call the Hurewicz dichotomy) for ∑11j subsets of the generalized Baire space κκ for a given uncountable cardinal κ with κ = κ<κ. We show that the statement that this dichotomy holds at all uncountable regular cardinals is consistent with the axioms of ZFC together with GCH and large cardinal axioms. In contrast, we show that the dichotomy fails at all uncountable regular cardinals after we add a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the κ-perfect set property, the κ-Miller measurability, and the κ-Sacks measurability. |