Compactness in the fine and related topologies

Let X be a Tychonoff space, Y a metrizable space and C(X,Y) be the space of continuous functions from X to Y. For a paracompact, locally hemicompact k-space X we characterize compact subsets of C(X,Y) topologized with the fine, graph and Krikorian topologies. Our results concerning compactness in the fine topology greatly generalized those of Spring [Topology Appl. 18 (1984) 87].