The Hurewicz dichotomy for generalized Baire spaces

Abstract

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 ω ω. We consider the analogous statement (which we call Hurewicz dichotomy) for Σ 1 1 subsets of the generalized Baire space κ κ 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.

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

“…Moreover, we do not know if the two cases in the perfect set dichotomy in Theorem 4.9 are mutually exclusive unless 2 ω < 2 ω1 . This is related to the question over which models it is possible to add perfect subsets of the ground model (see [19,Lemma 6.2] and [34]). Question 7.4.…”

Questions on generalised Baire spaces

“…In , Schlicht constructed a model where all projective sets satisfy the generalised perfect set property. The related Hurewicz dichotomy was first studied in . It consistently fails for closed sets and consistently holds for sets.…”

“…Moreover, it is well‐known that 2 κ and are homeomorphic if and only if κ is not a weakly compact cardinal (cf. [, Theorem 1] and [, Corollary 2.3]). Therefore, if κ is not weakly compact, the perfect set property implies the Hurewicz dichotomy, and hence it consistently holds for all projective sets.…”

“…Further background . Note that the notions of “κ‐Miller measurability”, “κ‐Sacks measurability” and “κ‐Silver measurability” considered in and are potential candidates for such properties; however, they are not generated by ‐closed forcing notions on .…”

“…Further background. Note that the notions of "κ-Miller measurability", "κ-Sacks measurability" and "κ-Silver measurability" considered in [48] and [44] are potential candidates for such properties; however, they are not generated by <κ-closed forcing notions on κ κ . Question 3.34 (Friedman, Laguzzi) Is there a version of generalised Silver forcing which is <κ-closed and such that for the corresponding regularity property, one can force that all generalised projective sets are regular?…”