Functorial uniformization of topological spaces

AbstractLet T be the forgetful functor from uniform spaces to completely regular topological spaces. We study T-sections, i.e. functors right inverse to T. We develop as tool the notion of spanning a T-section by a class of uniform spaces, and the order-dual notion of cospanning. Coarsest and finest uniform bireflectors and coreflectors associated with a T-section are characterized. Certain effects of the uniform completion reflector on a T-section are expressed in terms of the associated bireflectors