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Fuzzy equivalence and the resulting topology. (English) Zbl 0776.04002

This paper presents a construction for a topology on a set \(E\) on which a fuzzy equivalence is defined. A fuzzy equivalence relation is a generalization of the usual notion of equivalence relation where transitivity has been extended in a way consistent with the theory of fuzzy sets. The authors then use the equivalence relation to construct a Kuratowski closure operator on \(E\). If the triangle function that supports transitivity is dense, then the resulting topology corresponds in a natural way to the underlying equivalence relation. The authors then give a connection between this construction and Poincaré’s conception of the physical continuum.

MSC:

03E72 Theory of fuzzy sets, etc.
54A05 Topological spaces and generalizations (closure spaces, etc.)
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[1] Chevallard, Y.; Johsua, M. A., La Notion de Distance: Un exemple de la transposition didactique, Rech. Didactique Math., 3, 2, 159-239 (1982)
[2] Drossos, C. A.; Markakis, G., Boolean fuzzy sets, Fuzzy Sets and Systems, 46, 81-95 (1992) · Zbl 0760.03016
[3] Fréchet, M., Sur quelques points du calcul fonctionnel, Rent. Circ. Matem. Palermo, XXII, 1-74 (1906) · JFM 37.0348.02
[4] Fritsche, R., Topologies for probabilistic metric spaces, Fund. Math., 72, 7-16 (1971) · Zbl 0219.54045
[5] Menger, K., Statistical metrics, (Proc. Nat. Acad. Sci. U.S.A., 28 (1942)), 535-537 · Zbl 0063.03886
[6] Menger, K., Probabilistic theories of relations, (Proc. Nat. Acad. Sci. U.S.A., 37 (1951)), 178-180 · Zbl 0042.37103
[7] Ovchinnikov, S. V., Structure of fuzzy binary relations, Fuzzy Sets and Systems, 6, 169-195 (1981) · Zbl 0464.04004
[8] Poincaré, H., Science and Hypothesis (1952), The Science Press: The Science Press Lancaster, PA, republished by Dover · Zbl 0049.29106
[9] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific J. Math., 10, 313-334 (1960) · Zbl 0091.29801
[10] Schweizer, B.; Sklar, A., Associative functions and statistic triangle inequalities, Publ. Math. Debrecen, 8, 169-186 (1961) · Zbl 0107.12203
[11] Tardiff, R. M., Topologies for probabilistic metric spaces, Pacific J. Math., 65, 1, 233-251 (1976) · Zbl 0337.54004
[12] Yeh, R. T.; Bang, S. Y., Fuzzy relations, fuzzy graphs and their applications, (Zadeh, L. A.; Fu, K. S.; Tanaka, K.; Shimura, M., Fuzzy Sets and their Applications to Cognitive and Decision Processes (1975), Academic Press: Academic Press New York) · Zbl 0315.68069
[13] Zadeh, L. A., Similarity relations and fuzzy ordering, Inform. Sci., 3, 177-200 (1971) · Zbl 0218.02058
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