COUNTABLY COMPACT FIRST COUNTABLE SUBSPACES OF ORDINALS HAVE THE SOKOLOV PROPERTY

A space X is Sokolov if for any sequence {F(n) : n is an element of N} where F(n) is a closed subset of X(n) for every n is an element of N, there exists a continuous map f : X -> X such that nw (f(X)) <= omega and f(n) (F(n)) subset of F(n) for all n is an element of 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 is an element of omega. Another consequence of the above results is the existence of a fi rst countable Sokolov space of cardinality greater than c.