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Separation axioms for completeness and total boundedness in fuzzy pseudo metric spaces. (English) Zbl 0589.54006
The paper mainly aims at the introduction of fuzzy pseudo-metric on a set X along with the pseudo-metric topology and continues the author’s study in ibid. 86, 74-95 (1982; Zbl 0501.54003). For discussing the properties of the topology, the author introduces topological properties like separation axioms, separability, axioms of countability and a compactness-like concept which the author calls m-compactness. A fuzzy metric on X, Cauchy sequences, m-convergence and subsequently completeness along with total boundedness are also introduced. For Urysohn’s lemma the author uses the fuzzy unit interval with some variation of its previous version. The expected relations between the separation axioms are valid. The equivalence of separability and 2nd countability in a fuzzy pseudo-metric space is established. The most interesting result is a version of Baire’s theorem and the equivalence of m-compactness with total boundedness and completeness. The author also introduces contraction mappings along with a fixed-point theorem.
Reviewer: S.Ganguly

MSC:
54A40 Fuzzy topology
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54E15 Uniform structures and generalizations
54D30 Compactness
54E52 Baire category, Baire spaces
Citations:
Zbl 0501.54003
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References:
[1] Deng, Z., Neighborhoods in fuzzy topological spaces, J. hunan univ., 3, 1-11, (1980)
[2] Hutton, B., Normality in fuzzy topological spaces, J. math. anal. appl., 50, 74-97, (1975) · Zbl 0297.54003
[3] Deng, Z., Fuzzy pseudo metric spaces, J. math. anal. appl., 86, 74-95, (1982) · Zbl 0501.54003
[4] Erceg, M.A., Metric spaces in fuzzy set theory, J. math. anal. appl., 69, 205-236, (1979) · Zbl 0409.54007
[5] Pervin, W.T., Foundations of general topology, (1964), Academic Press New York · Zbl 0117.39701
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