Dual uniformities in function spaces over uniform continuity

Abstract

Abstract The notion of dual uniformity is introduced on U C ( Y , Z ) UC(Y,Z) , the uniform space of uniformly continuous mappings between Y Y and Z Z , where ( Y , V ) (Y,{\mathcal{V}}) and ( Z , U ) (Z,{\mathcal{U}}) are two uniform spaces. It is shown that a function space uniformity on U C ( Y , Z ) UC(Y,Z) is admissible (resp. splitting) if and only if its dual uniformity on U Z ( Y ) = { f 2 − 1 ( U ) ∣ f ∈ U C ( Y , Z ) , U ∈ U } {{\mathcal{U}}}_{Z}(Y)=\{{f}_{2}^{-1}(U)\hspace{0.33em}| \hspace{0.33em}f\in UC(Y,Z),U\in {\mathcal{U}}\} is admissible (resp. splitting). It is also shown that a uniformity on U Z ( Y ) {{\mathcal{U}}}_{Z}(Y) is admissible (resp. splitting) if and only if its dual uniformity on U C ( Y , Z ) UC(Y,Z) is admissible (resp. splitting). Using duality theorems, it is also proved that the greatest splitting uniformity and the greatest splitting family open uniformity exist on U Z ( Y ) {{\mathcal{U}}}_{Z}(Y) and U C ( Y , Z ) UC(Y,Z) , respectively, and these two uniformities are mutually dual splitting uniformities.