Let X and Y be topological spaces and F (X, Y ) the set of all functions from X into Y . We study various quasi-uniform convergence topologies UA (A ⊆ P (X)) on F (X, Y ) and their comparison in the setting of Y a quasi-uniform space. Further, we study UA-closedness and right Kcompleteness properties of certain subspaces of generalized continuous functions in F (X, Y ) in the case of Y a locally symmetric quasi-uniform space or a locally uniform space.2010 MSC: 54C35; 54E15; 54C08.Keywords: quasi-uniform space; topology of quasi-uniform convergence on a family of sets; locally uniform spaces, right K-completeness; quasi-continuous functions; somewhat continuous functions.