WEAK AND SUPPORT-OPEN TOPOLOGIES ON C(X)

This paper studies the weak topology w on the set C(X) of all continuous real valued functions on X, with respect to the dual of C-k(X) where C-k(X) has the compact-open topology. A related set-open topology s on C(X) is introduced using sets that are supports of the continuous linear functionals in the dual of C-k(X). It is shown that s is finer than w, and that under certain conditions if a set-open topology on C(X) is finer than s, then the dual of this function space is equal to the dual of C-k(X). Characterizations of metrizability and completeness of these function spaces are given in terms of topological properties of X. Also an Ascoli theorem is established for C-s(X), and from this it follows that C-k(X) is the k-extension of C-s(X) for certain X.