677 research outputs found
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
Baire spaces and infinite games
It is well known that if the nonempty player of the Banach-Mazur game has a
winning strategy on a space, then that space is Baire in all powers even in the
box topology. The converse of this implication may be true also: We know of no
consistency result to the contrary. In this paper we establish the consistency
of the converse relative to the consistency of the existence of a proper class
of measurable cardinals.Comment: 21 page
- …