Coincidence of function space topologies

We consider conditions under which the compact-open, Isbell, or natural topologies on the set of continuous real-valued functions on a space coincide. To this end, we consider the natural topology on the set of upper semicontinuous set-valued functions and give a concrete description of its open sets. This description allows us to give a number examples of function spaces where the compact-open, Isbell, and natural topologies do or do not agree. We show that R -concordance and local compactness coincide for metric spaces. We find the first example of an R -harmonic non-locally compact space. Also, under some set-theoretical hypotheses, we find the first example of an R -concordant non- R -harmonic space.