On Souslin sets and embeddings in integer-valued function spaces on ω1

Abstract

Let Σ( N ℵ 1 ) be the subspace of the t% 1-product of natural numbers N ℵ 1 , consisting of functions with countable support. We prove that for any uncountable Souslin set A in gS( N ℵ 1 ) , either A contains a Cantor set, or a copy of ω 1 (the space of countable ordinals) or else A can be well-ordered in type ω 1 so that all initial segments are closed (Theorem 1.1). We give also a more refined version of this result (Theorem 1.2). In particular, we demonstrate non-effectiveness of some selections from natural “layers” in Σ( N ℵ 1 ) , extending some ideas of A.H. Stone concerning Borel theory in nonseparable metrizable spaces. Connections of this subject to classical Lusin's constituents are also discussed. In another direction, we indicate (Corollary 1.3) a locally countable non-Souslin set in N ℵ 1 (witnessing poor covering properties of N ℵ 1 and answering a question by Kemoto and Yajima), and we find a closed perfectly normal subspace of N ℵ 1 which is not a countable union of closed subsets with finite covering dimension.