Abstract
Let X and Y be two topological spaces, F (X; Y) the set of all functions from X into Y and C (X; Y) the set of all continuous functions in F (X; Y). In the case of Y - (Y; U), a uniform space, various uniform convergence topologies (such as U-X; U-k; U-p) on F (X; Y) and C (X; Y) were systematically studied by Kelley ([7], Chapter 7). Let S (X; Y) (resp. P (X; Y), L (X; Y)) denote the subspace of C (X; Y) consisting of functions of all strongly continuous (resp. perfectly continuous, cl-supercontinuous) functions from X into Y. In the setting of (Y; U) a quasi-uniform space, we extend some recent results of Kohli and Singh [9] and Kohli and Aggarwal [8] on U-X -closedness and completeness of L (X; Y) in C (X; Y), and also on U-p-closedness and compactness of L (X; Y) in C (X; Y). In doing so, we fi rst need to extend several supporting results of Kelley ([7], Chapter 7) on joint continuity, equicontinuity and even continuity from the setting of uniform to quasi-uniform spaces.