Countably compact first countable subspaces of ordinals have the Sokolov property

Abstract

A space X is Sokolov if for any sequence {Fn : n ∈ ℕ} where Fn is a closed subset of Xn for every n ∈ ℕ, there exists a continuous map f : X → X such that nw(f(X)) ≤ ω and fn(F n ) ⊂ Fn for all n ∈ ℕ. We prove that if X is a first countable countably compact subspace of an ordinal then X is a Sokolov space and CP(X) is a D-space; this answers a question of Buzyakova. Thus, for any first countable countably compact subspace X of an ordinal, the iterated function space Cp , 2n +1(X) is Lindelöf for any n ∈ ω Another consequence of the above results is the existence of a first countable Sokolov space of cardinality greater than c.