⊤-uniform convergence spaces

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