Topologies on the space of continuous functions

Let X and Y be Tychonoff spaces and C(X, Y) be the space of all continuous functions from X to Y. The coincidence of the fine topology with other function space topologies on C(X, Y) is discussed. Also cardinal invariants of the fine topology on C(X, R ) , where R is the space of reals, are studied. To answer some questions of Di Maio and Naimpally (1992) other function space topologies are investigated, namely, Krikorian topology, open-cover topology, graph topology, topology of uniform convergence, proximal graph topology.