  For a topological space X, let (RX)s := (RX,Ts) be the cartesian product of
|X| copies of the real line R with the topology of the uniform convergence on
separable subsets of X. In this article we analyze the subspace C(X) of (RX)s
of all real-valued continuous functions on X, denoted by Cs(X). We determine
when Cs(X) is dense and when is closed in (RX)s, and we obtain some results
about the Baire property in Cs(X). Finally, we determine the cellularity of
Cs([0,{\alpha}]) where [0,{\alpha}] is the space of ordinal numbers belonging
to {\alpha} + 1 with its usual order topology.
