Countably compact first countable subspaces of ordinals have the Sokolov property

Abstract

A space X is Sokolov if for any sequence {F_n : n ∈ ℕ} where F_n is a closed subset of X_n for every n ∈ ℕ, there exists a continuous map f : X → X such that nw(f(X)) ≤ ω and f_n(F _n ) ⊂ F_n 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 C_P(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 C_p , 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.