Functional equicontinuity and uniformities in topological groups

A set H of continuous mappings from a topological space X to a uniform space (Y,U) is said to be functionally equicontinuous (at x∈X) if the set {f∘h: h∈H} is equicontinuous (at x) for each bounded real-valued uniformly continuous function f on (Y,U). In this paper, information about this concept is given. The relation between equicontinuity (at x∈X) and functional equicontinuity (at x∈X) is examined in detail. The main result asserts that for every X belonging to a wide class C of topological spaces (including all quasi-kR-spaces), any set of continuous mappings from X to any uniform space Y which is functionally equicontinuous is in fact equicontinuous. Applications to topological groups of the general results are given in the last section. In particular, the main result is applied to solve positively in the class C the problem of the equality [FSIN]=[SIN] raised by Itzkowitz.