We extend Solovay's theorem about definable subsets of the Baire space to the
generalized Baire space λλ, where λ is an uncountable
cardinal with λ<λ=λ. In the first main theorem, we show
that that the perfect set property for all subsets of λλ
that are definable from elements of λOrd is consistent
relative to the existence of an inaccessible cardinal above λ. In the
second main theorem, we introduce a Banach-Mazur type game of length λ
and show that the determinacy of this game, for all subsets of
λλ that are definable from elements of
λOrd as winning conditions, is consistent relative to the
existence of an inaccessible cardinal above λ. We further obtain some
related results about definable functions on λλ and
consequences of resurrection axioms for definable subsets of
λλ