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

Let Σ(Nℵ1) be the subspace of the t%1-product of natural numbers Nℵ1, consisting of functions with countable support. We prove that for any uncountable Souslin set A in gS(Nℵ1), either A contains a Cantor set, or a copy of ω1 (the space of countable ordinals) or else A can be well-ordered in type ω1 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(Nℵ1), extending some ideas of A.H. Stone concerning Borel theory in nonseparable metrizable spaces. Connections of this subject to classical Lusinʹs constituents are also discussed. In another direction, we indicate (Corollary 1.3) a locally countable non-Souslin set in Nℵ1 (witnessing poor covering properties of Nℵ1 and answering a question by Kemoto and Yajima), and we find a closed perfectly normal subspace of Nℵ1 which is not a countable union of closed subsets with finite covering dimension.