Countably compact first countable subspaces of ordinals have the Sokolov property.

A space X is Sokolov if for any sequence {Fn : n ∈ } where Fn is a closed subset of Xn for every n ∈ , there exists a continuous map f : X → X such that nw(f(X)) ≤ ω and fn(Fn) ⊂ Fn for all 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 Lindelof for any n ∈ ω Another consequence of the above results is the existence of a first countable Sokolov space of cardinality greater than c. [ABSTRACT FROM AUTHOR]