Completeness in probabilistic quasi-uniform spaces. (English) Zbl 1423.54028
Summary: In this paper, we give a kind of Cauchy completeness in probabilistic quasi-uniform space by using pair \(\top\)-filters, and prove that this kind of Cauchy completeness is equivalent to the \(\top\)-completeness of induced probabilistic uniform space introduced by Höhle. We also give a characterization of Cauchy completeness by Lawvere completeness. Then, we study the completion of probabilistic (quasi-)uniform space and show that each \(T_0\) separated probabilistic quasi-uniform space has a \(T_0\) Cauchy completion. Finally, in probabilistic quasi-metric space, we study the relationship between the completeness of induced quasi-uniform space and the completeness of probabilistic quasi-metric space.
MSC:
| 54A40 | Fuzzy topology |
| 54D10 | Lower separation axioms (\(T_0\)–\(T_3\), etc.) |
| 54E15 | Uniform structures and generalizations |
| 54E70 | Probabilistic metric spaces |
Keywords:
probabilistic quasi-uniformity; pair \(\top\)-filter; Cauchy completeness; Lawvere completeness; probabilistic quasi-metricReferences:
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