Semi-uniform convergence topologies on function spaces

We study semi-uniform convergence topologies on the function space F (X, Y ), where X is a topological space and (Y, U ) is a semi-uniform space.Particular focus is given to the compact convergence topology U k on C(X, Y ).We characterize the conditions under which U k coincides with the compact-open topology and establish criteria for its semi-pseudometrizability and completeness in terms of hemicompactness of X and the structure of Y .These results offer insight into the relationship between semi-uniform convergence and classical function space topologies.