Lindelöf property and the iterated continuous function spaces

We give an example of a compact space X whose iterated continuous function spaces Cp(X), CpCp(X), . . . are Lindelöf, but X is not a Corson compactum. This solves a problem of Gul’ko (Problem 1052 in [11]). We also provide a theorem concerning the Lindelöf property in the function spaces Cp(X) on compact scattered spaces with the ω1th derived set empty, improving some earlier results of Pol [12] in this direction. 1. Notation and terminology. Our terminology follows Arkhangel’skĭı [1]. Given a topological space X, we denote by Cp(X) the space of real-valued continuous functions, equipped with the topology of pointwise convergence, and CpCp(X), CpCpCp(X), . . . are the iterated continuous function spaces. We denote by D the discrete two-point space {0, 1} and Cp(X,D) = {f ∈ Cp(X) : f(X) ⊂ D}. We denote by ω1 the set of all countable ordinals. A set A ⊂ ω1 is stationary if it intersects each closed set (in the order topology), unbounded in ω1; we call A bistationary if both A and ω1 \A are stationary. For information concerning stationary sets needed in this paper we refer the reader to Jech [7], or Fleissner [3]. A topological spaceX is א0-monolithic if for any countable subset A ⊂ X the closure cl(A) has a countable network (see [1; Ch. II, §6]); a network for a space Y is a family of sets such that each open set is the union of some subfamily of this family. A topological space X has countable tightness if for any x in cl(A) there is an at most countable subset B ⊂ A with x ∈ cl(B) (see [1]). A compact space X is a Corson compactum if X can be embedded in the subspace of the Tikhonov product R of the real line consisting of functions vanishing at all but countably many points in Γ (see [1], [9]).