The uniform convergence topology on separable subsets

For a topological space X, let be the cartesian product of copies of the real line with the topology of the uniform convergence on separable subsets of X. In this article we analyze the subspace of of all real-valued continuous functions on X, denoted by. We determine when is dense and when is closed in, and we obtain some results about the Baire property in. Finally, we determine the cellularity of where is the space of ordinal numbers belonging to with its usual order topology. ©The authors ©Topology and its Applications ©Elsevier.