Dual uniformities in function spaces over uniform continuity (Q6049684)

Let \((Y,\mathcal{V})\) and \((Z,\mathcal{U})\) be uniform spaces and let \(UC(Y,Z)\) be the set of all uniformly continuous mappings from \(Y\) to \(Z\). In the paper under review, the authors introduce the concept of dual uniformity on \(UC(Y,Z)\). They prove that a uniformity on \(UC(Y,Z)\) is admissible (resp. splitting) if and only if its dual uniformity on \(\mathcal{U}_{Z}(Y)\) is admissible (resp. splitting). Also it is proved that a uniformity on \(\mathcal{U}_{Z}(Y)\) is admissible (resp. splitting) if and only if its dual uniformity on \(UC(Y,Z)\) is admissible (resp. splitting). The authors show the existence of the greatest splitting uniformity on \(\mathcal{U}_{Z}(Y)\) as well as the greatest splitting family open uniformity on \(UC(Y,Z)\) and prove that these two uniformities are mutually dual splitting uniformities.