⊤-uniform convergence spaces

We show, for a commutative and integral quantale, that the recently introduced  category of \top-uniform convergence spaces is a topological category which possesses natural function spaces, making it Cartesian closed. Furthermore, as two important examples for \top-uniform convergence spaces, we study probabilistic uniform spaces and quantale-valued metric spaces. The underlying \top-convergence spaces are also described and it is shown that in the case of a probabilistic uniform space this \top-convergence is the convergence of a fuzzy topology with conical neighbourhood filters. Finally it is shown that the category of \top-uniform convergence spaces can be embedded into the category of stratified lattice-valued uniform convergence spaces as a reflective subcategory.

Keywords:

topology , top-filter , uniform convergence space , lattice-valued uniform convergence space , probabilistic uniform space , quantale-valued metric space