Selection principles in function spaces with the compact-open topology

Abstract

For a Tychonoff space X, we denote by Ck(X) the space of all real-valued continuous functions on X with the compact-open topology. A subset A ? X is said to be sequentially dense in X if every point of X is the limit of a convergent sequence in A. In this paper, the following properties for Ck(X) are considered. S1(S,S)=> Sfin(S,S) => Sfin(S,D) <=S1(S,D) S1(D,S) => Sfin(D,S) => Sfin(D,D) <= S1(D,D) For example, a space Ck(X) satisfies S1(S,D) (resp., Sfin(S,D)) if whenever (Sn : n ? N) is a sequence of sequentially dense subsets of Ck(X), one can take points fn ? Sn (resp., finite Fn ? Sn) such that {fn : n ? N} (resp.,U {Fn : n ? Ng) is dense in Ck(X). Other properties are defined similarly. In [22], we obtained characterizations these selection properties for Cp(X). In this paper, we give characterizations for Ck(X).