Abstract
The paper deals with monadic as well as monadic-free topological notions. For defining these monadic-free notions the notion of basic triple Φ is introduced. A lot of monadic-free topological notions are presented, for instance that of Φ-convergence structure, Φ-hull operator and Φ-uniform structure. By means of a generalized metric, e.g. a probabilistic metric, and the general notion of Φ-zero approach introduced in this paper, a Φ-uniform structure is generated. In case of a fuzzy metric the related Φ-uniform structure defines in a canonic way a fuzzy topology which is used for developing a fuzzy analysis and fuzzy calculus.
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References
P. Eklund and W. Gähler, Generalized Cauchy spaces, Math. Nachr., 147 (1990), 219–233.
P. Eklund and W. Gähler, Fuzzy filter functors and convergence, in: Applications of Category Theory to Fuzzy Subsets, Kluwer Academic Publishers (Dordrecht, 1992), pp. 109–136.
S. Gähler and W. Gähler, Fuzzy real numbers, Fuzzy Sets and Systems, 66 (1994), 137–158.
W. Gähler, Monadic topology — a new concept of generalized topology, in: Recent Developments of General Topology and its Applications, International Conference in Memory of Felix Hausdorff (1868–1942), Math. Research 67, Akademie Verlag (Berlin, 1992), pp. 118–123.
W. Gähler, Convergence, Fuzzy Sets and Systems, 73 (1995), 97–129.
W. Gähler, The general fuzzy filter approach to fuzzy topology, I, Fuzzy Sets and Systems, 76 (1995), 205–224.
W. Gähler, The general fuzzy filter approach to fuzzy topology, II, Fuzzy Sets and Systems, 76 (1995), 225–246.
W. Gähler, On fuzzy analysis and fuzzy calculus, J. Egypt. Math. Soc., 7 (1999), 1–16.
W. Gähler, Monadic convergence structures, in: Serninarberichte aus dem Fachbereich Mathematik, Fernuniversität Hagen, 67 (1999), pp. 111–130.
W. Gähler, F. Bayoumi, A. Kandil and A. Nouh, The theory of global fuzzy neighborhood structures, Part III, Fuzzy uniform structures, Fuzzy Sets and Systems, 98 (1998), 175–199.
W. Gähler and S. Gähler, Contributions to fuzzy analysis, Fuzzy Sets and Systems, 105 (1999), 201–224.
U. Höhle, Locales and L-topologies, Mathematik-Arbeitspapiere 48. Univ. Bremen (1997), 223–250.
U. Höhle and A. Šostak, Axiomatics of fixed-basis fuzzy topologies, in: Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers (Dordrecht, Boston, 1999), pp. 123–273.
D. C. Kent, On convergence groups and convergence uniformities, Fund. Math., 60 (1967), 213–222.
E. Trillas, Sobre distancias aleatorias, Actas R.A.M.E. (C.S.I.C), Santiago de Compostela (1967).
B. Schweitzer and A. Sklar, Probabilistic Metric Spaces, North Holland (1983).
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Gähler, W. General Topology — the Monadic Case, Examples, Applications. Acta Mathematica Hungarica 88, 279–290 (2000). https://doi.org/10.1023/A:1026723922622
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DOI: https://doi.org/10.1023/A:1026723922622