The Hurewicz dichotomy for generalized Baire spaces

Abstract

By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space $X$ is covered by a $K_{\sigma }$ subset of $X$ if and only if it does not contain a closed-in-$X$ subset homeomorphic to the Baire space ${}^{\omega }\omega$. We consider the analogous statement (which we call Hurewicz dichotomy) for ${\Sigma }_1^1$ subsets of the generalized Baire space ${}^{\kappa }\kappa$ for a given uncountable cardinal $\kappa$ with $\kappa ={\kappa }^{<\kappa }$, and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the Generalized Continuum Hypothesis (GCH) holds, then there is a cardinal preserving class-forcing extension in which the Hurewicz dichotomy for ${\Sigma }_1^1$ subsets of ${}^{\kappa }\kappa$ holds at all uncountable regular cardinals $\kappa$, while strongly unfoldable and supercompact cardinals are preserved. On the other hand, in the constructible universe L the dichotomy for ${\Sigma }_1^1$ sets fails at all uncountable regular cardinals, and the same happens in any generic extension obtained by adding a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the $\kappa$-perfect set property, the $\kappa$-Miller measurability, and the $\kappa$-Sacks measurability.

33 pages, final version

References in corpus (1)

• Generalized Silver and Miller measurability

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