Dual uniformities in function spaces over uniform continuity

Abstract

The notion of dual uniformity is introduced on UC(Y,Z)UC(Y,Z), the uniform space of uniformly continuous mappings between YY and ZZ, 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 UC(Y,Z)UC(Y,Z) is admissible (resp. splitting) if and only if its dual uniformity on UZ(Y)={f2−1(U)∣f∈UC(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 UZ(Y){{\mathcal{U}}}_{Z}(Y) is admissible (resp. splitting) if and only if its dual uniformity on UC(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 UZ(Y){{\mathcal{U}}}_{Z}(Y) and UC(Y,Z)UC(Y,Z), respectively, and these two uniformities are mutually dual splitting uniformities.

Keywords

• dual uniformity
• uniform space
• function space
• splittingness
• admissibility
• 54c35
• 54a20