10 research outputs found
The Hurewicz dichotomy for generalized Baire spaces
By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic
subset of a Polish space is covered by a subset of if and
only if it does not contain a closed-in- subset homeomorphic to the Baire
space . We consider the analogous statement (which we call
Hurewicz dichotomy) for subsets of the generalized Baire space
for a given uncountable cardinal with
, 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
subsets of holds at all uncountable regular
cardinals , while strongly unfoldable and supercompact cardinals are
preserved. On the other hand, in the constructible universe L the dichotomy for
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
-perfect set property, the -Miller measurability, and the
-Sacks measurability.Comment: 33 pages, final versio
Perfect subsets of generalized Baire spaces and long games
We extend Solovay's theorem about definable subsets of the Baire space to the
generalized Baire space , where is an uncountable
cardinal with . In the first main theorem, we show
that that the perfect set property for all subsets of
that are definable from elements of is consistent
relative to the existence of an inaccessible cardinal above . In the
second main theorem, we introduce a Banach-Mazur type game of length
and show that the determinacy of this game, for all subsets of
that are definable from elements of
as winning conditions, is consistent relative to the
existence of an inaccessible cardinal above . We further obtain some
related results about definable functions on and
consequences of resurrection axioms for definable subsets of
Generalized Silver and Miller measurability
We present some results about the burgeoning research area concerning set theory of the "kappa-reals". We focus on some notions of measurability coming from generalizations of Silver and Miller trees. We present analogies and mostly differences from the classical setting. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei
Generalized Silver and Miller measurability
1 We present some results about the burgeoning research area concern-ing set theory of the “κ-reals”. We focus on some notions of measurability coming from generalizations of Silver and Miller trees. We present analo-gies and mostly differences from the classical setting. 1 Introduction and basic definitions The study of the generalized version of the Baire space κκ and Cantor space 2κ, for κ uncountable regular cardinal, is a burgeoning research area, which intersects both the generalized descriptive set theory and the set theory of the “κ-reals”, where we refer to the elements of κκ and 2κ as κ-reals