Uniformity of uniform convergence on the family of sets

We prove that for every Hausdorff space X and any uniform quadra space (Y,U) the topology on C(X,Y) induced by the uniformity U| of uniform convergence on the saturation family L coincides with the set-open topology on C(X,Y). In particular, for every pseudocompact space X and any metrizable topological vector space Y with uniform U the topology on C(X,Y) induced by the uniformity U| of uniform convergence coincides with the C-compact-open topology on C(X,Y), and depends only on the topology induced on Y by the uniformity U. It is also shown that in the class closed-homogeneous complete uniform spaces Y necessary condition for coincidence of topologies is Y-compactness of elements of family L.

💡 Research Summary

The paper investigates the relationship between two natural topologies on the space C(X,Y) of continuous maps from a Hausdorff space X into a uniform space (Y,U). The first topology is induced by the uniformity of uniform convergence on a prescribed family L of subsets of X; the second is the set‑open (or set‑wise) topology generated by subbasic sets of the form \