Functorial admissible quasi-uniformities on topological spaces

Abstract

Let T denote the forgetful functor from the category Quu of quasi-uniform spaces and quasi-uniformly continuous maps to the category Top of topological spaces and continuous maps. In the first part of this paper we develop a general method to construct nontransitive admissible quasi-uniformities on topological spaces. We use this technique to answer a question of G.C.L. Brümmer in the negative by exhibiting a nontransitive T-section that is coarser than the (transitive) locally finite covering quasi-uniformity. In the second part of this paper we show that the Pervin functor restricted to the category Haus of Hausdorff spaces and continuous maps is the coarsest functor that puts compatible quasi-uniformities on the Hausdorff spaces.