Sequential Completeness for ⊤-Quasi-Uniform Spaces and a Fixed Point Theorem

Abstract

We define sequential completeness for ⊤-quasi-uniform spaces using Cauchy pair ⊤-sequences. We show that completeness implies sequential completeness and that for ⊤-uniform spaces with countable ⊤-uniform bases, completeness and sequential completeness are equivalent. As an illustration of the applicability of the concept, we give a fixed point theorem for certain contractive self-mappings in a ⊤-uniform space. This result yields, as a special case, a fixed point theorem for probabilistic metric spaces.