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Basic structures of fuzzy topology

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Literature Cited

  1. A. V. Arkhangel'skii and V. I. Ponomarev,Foundations of General Topology in Problems and Exercises [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  2. N. Bourbaki,General Topology. Basic Structures [Russian translation], Nauka, Moscow (1968).

    Google Scholar 

  3. Z. B. Diskin, “Fuzzy predicates on fuzzy spaces,” In:Topological Spaces and Mappings [in Russian], Latvian University Press, Riga (1985), pp. 59–70.

    Google Scholar 

  4. V. A. Efremovich, “Infinitesimal spaces,”Dokl. Akad. Nauk SSSR,76, 341–343 (1951).

    Google Scholar 

  5. V. P. Salij, “BinaryL-relations,”Izv. Vuzov. Mat.,1, 133–145 (1965).

    MATH  Google Scholar 

  6. Yu. M. Smirnov, “On proximity spaces,”Mat. Sb.,31, 543–574 (1952).

    MATH  Google Scholar 

  7. A. P. Šostak, “Coreflectiveness in categories of fuzzy spaces,” In:Continuous Functions on Topological Spaces [in Russian], Latvian University Press, Riga (1986), pp. 159–165.

    Google Scholar 

  8. A. P. Šostak, “Coreflective subcategories of fuzzy topological spaces,” In:Theoretic and Applied Problems of Mathematics. Proc. of Tartu University Conference [in Russian], Tartu (1985), pp. 44–50.

  9. A. P. Šostak, “Separation axioms of fuzzy topological spaces,” In:Topological Spaces and Mappings [in Russian], Latvian University Press, Riga (1987), pp. 165–186.

    Google Scholar 

  10. A. P. Šostak, “Fuzzy topologies on spaces of probability measures,” In:Topological Spaces and Mappings [in Russian], Latvian University Press, Riga (1989), pp. 125–172.

    Google Scholar 

  11. R. Engelking,General Topology [in Russian], PWN, Warszawa (1977).

    Google Scholar 

  12. D. Adnadievic, “Separation properties of F-spaces,”Mat. Vestnik. 6, 1–8 (1982).

    Google Scholar 

  13. T. M. G. Ahsanullah and N. N. Morsi, “Invariant probabilistic metrizability of fuzzy neighborhood groups,”Fuzzy Sets Syst.,47, 233–245 (1992).

    Article  MathSciNet  Google Scholar 

  14. M. A. Amer and N. N. Morsi, “Characterization of some fuzzy neighborhood notions in probabilistic metric spaces,”Fuzzy Sets Syst.,51, 351–359 (1992).

    Article  MathSciNet  Google Scholar 

  15. G. Artico and R. Moresco, “Fuzzy proximities and totally bounded fuzzy uniformities,”J. Math. Anal. Appl.,99, 320–337 (1984).

    Article  MathSciNet  Google Scholar 

  16. G. Artico and R. Moresco, “On fuzzy metrizability,”J. Math. Anal. Appl.,107, 144–147 (1985).

    Article  MathSciNet  Google Scholar 

  17. G. Artico and R. Moresco, “Fuzzy proximities according with Lowen fuzzy uniformities,”Fuzzy Sets Syst.,21, 85–98 (1987).

    Article  MathSciNet  Google Scholar 

  18. G. Artico and R. Moresco, “Uniform properties of normal and compact fuzzy topological spaces,”J. Math. Anal. Appl.,132, 514–519 (1988).

    Article  MathSciNet  Google Scholar 

  19. G. Artico and R. Moresco, “Fuzzy uniformities induced by fuzzy proximities,”Fuzzy Sets Syst.,31, 111–121 (1989).

    Article  MathSciNet  Google Scholar 

  20. B. Banaschewskii, “The duality of distributive continuous lattices,”Can. J. Math.,32, 385–394 (1980).

    Google Scholar 

  21. G. Birkhoff,Lattice Theory, 3rd ed., Amer. Math. Soc., Providence, Rhode Island (1967).

    Google Scholar 

  22. M. H. Burton, “The relationship between a fuzzy uniformity and its family of α-level fuzzy uniformities,”Fuzzy Sets Syst.,54, 311–316 (1993).

    MATH  MathSciNet  Google Scholar 

  23. M. H. Burton, “Cauchy filters and prefilters,”Fuzzy Sets Syst.,54, 317–332 (1993).

    MATH  MathSciNet  Google Scholar 

  24. M. H. Burton, “Boundness in uniform spaces and fuzzy uniform spaces,”Fuzzy Sets Syst.,58, 195–208 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  25. M. H. Burton, “Completeness in fuzzy uniform spaces,”Quaest. Math.,16, 13–36 (1993).

    MATH  MathSciNet  Google Scholar 

  26. M. H. Burton, “Precompactness in fuzzy uniform spaces,”Quaest. Math.,16, 37–49 (1993).

    MATH  MathSciNet  Google Scholar 

  27. J. J. Chadwick, “A general form of compactness in fuzzy topological spaces,”J. Math. Anal. Appl.,162, 92–110 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  28. C. L. Chang, “Fuzzy topological spaces,”J. Math. Anal. Appl.,24, 182–190 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  29. F. T. Christoph, “Quotient fuzzy topology and local compactness,”J. Math. Anal. Appl.,57, 497–504 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  30. J. Coulon and J. L. Coulon, “Une notion de proximité floue,”Busefal,10, 17–34 (1982).

    Google Scholar 

  31. A. Csaszar,Foundations of General Topology, Pergamon Press (1963).

  32. C. De Mitri and E. Pascali, “Characterization of fuzzy topologies from neighborhoods of fuzzy points,”J. Math. Anal. Appl.,93, 1–14 (1983).

    MathSciNet  Google Scholar 

  33. C. H. Dowker and D. Papert, “Quotient frames and subspaces,”Proc. London Math. Soc.,16, 275–286 (1966).

    MathSciNet  Google Scholar 

  34. D. Dzhajanbajev and A. Šostak, “On completeness and precompactness spectra of fuzzy sets in fuzzy uniform spaces,”Comm. Math. Univ. Carol.,31, No. 1, 149–157 (1990).

    Google Scholar 

  35. D. Dzhajanbajev and A. Šostak, “On a fuzzy uniform structure,”Acta Comm. Univ. Tartu,940, 31–36 (1992).

    Google Scholar 

  36. P. Eklund, “Category theoretic properties of fuzzy topological spaces,”Fuzzy Sets and Syst.,19, 303–310 (1984).

    MathSciNet  Google Scholar 

  37. P. Eklund, “A comparison of lattice theoretic approaches to fuzzy topology,”Fuzzy Sets Syst.,19, 81–87 (1986).

    MATH  MathSciNet  Google Scholar 

  38. P. Eklund and W. Gähler, “Basic notions for fuzzy topology. Part I,”Fuzzy Sets Syst.,26, 333–356 (1988).

    Article  Google Scholar 

  39. P. Eklund and W. Gähler, “Basic notions for fuzzy topology. Part II,”Fuzzy Sets Syst.,27, 171–195 (1988).

    Article  Google Scholar 

  40. P. Eklund and W. Gähler, “Fuzzy filter functors and convergence,” In:Applications of Category Theory to Fuzzy Subsets (S. E. Rodabaugh, E. P. Klement, U. Hohle, eds.), Kluwer Acad. Publ. (1991), pp. 109–136.

  41. M. A. Erceg, “Functions, equivalence relations, quotient spaces and subsets in fuzzy set theory,”Fuzzy Sets Syst.,3, 75–92 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  42. R. Ertürk, “Separation axioms in fuzzy topologies characterized by bitopologies,”Fuzzy Sets Syst.,58, 209–216 (1993).

    Article  MATH  Google Scholar 

  43. Fan Taihe, “Product operations in the categories of molecular lattices,”Sci. Bull.,32, 505–570 (1987).

    Google Scholar 

  44. H. R. Fischer, “Limesraume,”Math. Ann.,137, 269–303 (1959).

    Article  MATH  MathSciNet  Google Scholar 

  45. D. H. Foster, “Fuzzy topological groups,”J. Math. Anal. Appl.,67, 549–564 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  46. T. E. Ganter, R. C. Steinlage, and R. H. Warren, “Compactness in fuzzy topological spaces,”J. Math. Anal. Appl.,62, 547–562 (1978).

    MathSciNet  Google Scholar 

  47. G. Gerla, “On fuzzy convergence,”Inf. Sci.,39, 269–284 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  48. M. H. Ghanim, E. E. Kerre, and A. S. Masshour, “Separation axioms, subspaces and sums in fuzzy topology,”J. Math. Anal. Appl.,102, 189–202 (1984).

    Article  MathSciNet  Google Scholar 

  49. M. H. Ghanim and A. S. Masshour, “Characterization of fuzzy topologies byS-neighborhood systems,”Fuzzy Sets Syst.,9, 211–212 (1983).

    Article  Google Scholar 

  50. G. Gierz, K. H. Hoffman, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott,A Compedium of Continuous Lattices, Springer Verlag, Berlin, Heidelberg, New York (1980).

    Google Scholar 

  51. J. A. Goguen, “\(\mathbb{L}\)-fuzzy sets,”J. Math. Anal. Appl.,18, 145–174 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  52. J. A. Goguen, “The fuzzy Tychonoff theorem,”J. Math. Anal. Appl.,43, 734–742 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  53. R. Goldblatt,Topos: the Categorical Analysis of Logic. Revised edition, North-Holland Publ. Co., Amsterdam (1981).

    Google Scholar 

  54. S. Gottwald, “Fuzzy points and local properties of fuzzy topological spaces,”Fuzzy Sets Syst.,5, 199–201 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  55. G. Gratzer,General Lattice Theory, Academie Verlag, Berlin (1978).

    Google Scholar 

  56. J. Gutierrez Garcia and M. A. de Prada Vicente,Fuzzy t-Uniformities, Univ. del Pais Basco, Bilbao, Spain (1993).

    Google Scholar 

  57. P. Hamburg, “Problemes actueles de la theorie des espaces topologiques flous,” In:Conf. Nationale de Geometrie si Topologie, Iasi (1984), pp. 215–224.

  58. H. Herrlich, “Are there convenient subcategories in Top?,”Topol. Appl.,15, 263–271 (1983).

    MATH  MathSciNet  Google Scholar 

  59. H. Herrlich, “Cartesian closed topological categories,”Math. Coll. Univ. Cape Town,9, 1–16 (1974).

    MATH  MathSciNet  Google Scholar 

  60. U. Höhle, “Probabilistsche topologien,”Manuscr. Math.,26, 223–245 (1978).

    Article  MATH  Google Scholar 

  61. U. Höhle, “Probabilistische Metriken auf der Menge der nicht negative Verteilungsfunctionen,”Aequat. Math.,18, 345–356 (1978).

    MATH  Google Scholar 

  62. U. Höhle, “Probabilistic uniformization of fuzzy topologies,”Fuzzy Sets Syst.,1, 311–332 (1978).

    MATH  Google Scholar 

  63. U. Höhle, “Upper semicontinuous fuzzy sets and applications,”J. Math. Anal.,78, 659–673 (1980).

    MATH  Google Scholar 

  64. U. Höhle, “Probabilistic metrization of fuzzy topologies,”Fuzzy Sets Syst.,8, 63–69 (1982).

    MATH  Google Scholar 

  65. U. Höhle, “Fuzzy topologies and topological space objects in a topos,”Fuzzy Sets Syst.,19, 299–304 (1986).

    MATH  Google Scholar 

  66. U. Höhle and Šostak, “A general theory of fuzzy topological spaces,”Fuzzy Sets Syst. (1994), in press.

  67. B. Hutton, “Normality in fuzzy topological spaces,”J. Math. Anal. Appl.,50, 74–79 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  68. B. Hutton, “Uniformities on fuzzy topological spaces,”J. Math. Anal. Appl.,58, 559–571 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  69. B. Hutton, “Products of fuzzy topological spaces,”Topol. Appl.,11, 59–67 (1980).

    MATH  MathSciNet  Google Scholar 

  70. B. Hutton and I. Reilly, “Separation axioms in fuzzy topological spaces,”Fuzzy Sets Syst.,3, 93–104 (1980).

    Article  MathSciNet  Google Scholar 

  71. P. T. Johnstone,Stone Spaces, Cambridge Univ. Press, Cambridge (1981).

    Google Scholar 

  72. A. K. Katsaras, “Fuzzy topological vector spaces. I,”Fuzzy Sets Syst.,6, 85–95 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  73. A. K. Katsaras, “Convergence of fuzzy filters in fuzzy topological spaces,”Bull. Math. Soc. Sci. Math. Roum.,27, 131–137 (1983).

    MATH  MathSciNet  Google Scholar 

  74. A. K. Katsaras, “Fuzzy proximity spaces,”J. Math. Anal. Appl.,68, 100–110 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  75. A. K. Katsaras, “On fuzzy proximity spaces,”J. Math. Anal. Appl.,75, 571–583 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  76. A. K. Katsaras, “Fuzzy proximities and fuzzy completely regular spaces,”J. Anal. St. Univ. Jasi,26, 31–41 (1980).

    MATH  MathSciNet  Google Scholar 

  77. A. K. Katsaras, “On fuzzy uniform spaces,”J. Math. Anal. Appl.,101, 97–113 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  78. A. K. Katsaras, “On fuzzy syntopogeneous structures,”Rev. Roum. Math. Pures Appl.,30, 419–431 (1985).

    MATH  MathSciNet  Google Scholar 

  79. A. K. Katsaras, “Fuzzy quasi-proximities and fuzzy quasi-uniformities,”Fuzzy Sets Syst.,27, 335–343 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  80. A. K. Katsaras, “Fuzzy neighborhood structures and fuzzy quasi-uniformities,”Fuzzy Sets Syst.,29, 187–200 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  81. A. K. Katsaras, “Totally bounded fuzzy syntopogeneous structures,”Fuzzy Sets Syst.,28, 91–105 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  82. A. K. Katsaras, “Fuzzy syntopogeneous structures compatible with Lowen fuzzy uniformities,”Fuzzy Sets Syst.,36, 375–393 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  83. A. K. Katsaras, “Operations on fuzzy syntopogeneous structures,”Fuzzy Sets Syst.,47, 199–218 (1991).

    MathSciNet  Google Scholar 

  84. A. K. Katsaras and C. G. Petalas, “A unified theory of fuzzy topologies, fuzzy proximities, and fuzzy uniformities,”Rev. Roum. Math. Pures Appl.,28, 845–856 (1983).

    MathSciNet  Google Scholar 

  85. A. K. Katsaras and C. G. Petalas, “On fuzzy syntopogeneous structures,”J. Math. Anal. Appl.,99, 219–236 (1984).

    Article  MathSciNet  Google Scholar 

  86. A. K. Katsaras and C. G. Petalas, “Fuzzy topologies and fuzzy quasi-uniformities,”Rad. Mat.,4, 15–29 (1988).

    MathSciNet  Google Scholar 

  87. J. L. Kelley,General Topology, New York (1955).

  88. E. E. Kerre and P. L. Ottoy, “On the different notions of neighborhood in fuzzy topological spaces,”Simon Steven,61, No. 2, 131–146 (1987).

    MathSciNet  Google Scholar 

  89. E. E. Kerre and P. L. Ottoy, “Lattice properties of neighborhood systems in Chang fuzzy topological spaces,”Fuzzy Sets Syst.,30, 205–214 (1987).

    MathSciNet  Google Scholar 

  90. A. J. Klein, “α-closure in fuzzy topology,”Rocky Mount. J. Math.,11, 553–560 (1981).

    MATH  Google Scholar 

  91. A. J. Klein, “Generating of fuzzy topologies with semi-closure operators,”Fuzzy Sets Syst.,9, 267–274 (1983).

    Article  MATH  Google Scholar 

  92. A. J. Klein, “When is a fuzzy topological space topological,”Rocky Mount. J. Math.,17, 15–18 (1987).

    MATH  Google Scholar 

  93. A. J. Klein, “Generating\(\mathbb{L}\)-fuzzy unit interval,”Fuzzy Sets Syst.,12, 271–279 (1984).

    Article  MATH  Google Scholar 

  94. W. Kotzé, “Quasi-coincidence and quasi-fuzzy Hausdorff,”J. Math. Anal. Appl.,116, 465–472 (1986).

    MATH  MathSciNet  Google Scholar 

  95. W. Kotzé, “The fuzzy Hausdorff property,”Glas. Mat.,20, 409–417 (1985).

    MATH  Google Scholar 

  96. W. Kotzé, “Lattice morphisms, sobriety and Uryson lemmas,” In:Appl. of Category Theory to Fuzzy Subsets (S. E. Rodabaugh, E. P. Klement, and U. Höhle, eds.), Kluwer Academic Publ. (1991). pp. 251–274.

  97. W. Kotzé and T. Kubiak, “Inserting\(\mathbb{L}\)-fuzzy real valued functions,”Math. Nachr.,164, 5–11 (1993).

    MathSciNet  Google Scholar 

  98. W. Kotzé and T. Kubiak, “Fuzzy topologies of Scott continuous functions and their relations to hypergraph functor,”Quaest. Math.,15, 175–187 (1992).

    Google Scholar 

  99. T. Kubiak, “Extending continuous\(\mathbb{L}\)-real functions,”Math. Jpn.,31, 875–887 (1986).

    MATH  MathSciNet  Google Scholar 

  100. T. Kubiak, “\(\mathbb{L}\)-fuzzy normal spaces and Tietze Extension Theorem,”J. Math. Anal. Appl.,125, 141–153 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  101. T. Kubiak,On Fuzzy Topologies (Ph. D. Thesis), A. Mickiewicz Univ., Poznan (1985).

    Google Scholar 

  102. T. Kubiak, “The topological modification of the\(\mathbb{L}\)-fuzzy unit interval,” In:Applications of Category Theory to Fuzzy Subsets (S. E. Rodabaugh, E. P. Klement, and U. Höhle, eds.), Kluwer Academic Publ. (1991), pp. 275–306.

  103. T. Kubiak and A. Šostak, “On\(\mathbb{L}\)-fuzzy topological spaces,”Fuzzy Sets Syst. (1994), in press.

  104. Liu Wang-jin, “Uniformity and proximity\(\mathbb{L}\)-fuzzy topological spaces,”Fuzzy Sets Syst.,30, 37–45 (1989).

    Article  Google Scholar 

  105. Liu Ying-ming, “On fuzzy convergence classes,”Fuzzy Sets Syst.,30, 47–51 (1989).

    Article  Google Scholar 

  106. E. Lowen and R. Lowen, “A topological universe extension of FTS,” In:Application of Category Theory to Fuzzy Subsets (S. E. Rodabaugh, E. P. Klement, and U. Höhle, eds.), Kluwer Academic Publ. (1991), pp. 153–176.

  107. R. Lowen, “Fuzzy topological spaces and fuzzy compactness,”J. Math. Anal. Appl.,56, 621–633 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  108. R. Lowen, “Initial and final fuzzy topologies and fuzzy Tychonoff theorem,”J. Math. Anal. Appl.,58, 11–21 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  109. R. Lowen, “A comparison of different compactness notions in fuzzy topological spaces,”J. Math. Anal. Appl.,64, 446–454 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  110. R. Lowen, “Convergence in fuzzy topological spaces,”General Topol. Appl.,10, 147–160 (1979).

    MATH  MathSciNet  Google Scholar 

  111. R. Lowen, “Compact Hausdorff fuzzy topological spaces are topological,”General Topol. Appl.,12, 65–74 (1981).

    MATH  MathSciNet  Google Scholar 

  112. R. Lowen, “Fuzzy neighborhood spaces,”Fuzzy Sets Syst.,7 165–189 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  113. R. Lowen, “Fuzzy uniform spaces,”J. Math. Anal. Appl.,82, 370–385 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  114. R. Lowen, “On (R(L), ⊕),”Fuzzy Sets Syst.,10, 203–209 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  115. R. Lowen, “Hyperspaces of fuzzy sets,”Fuzzy Sets Syst.,9, 287–312 (1983).

    Article  MathSciNet  Google Scholar 

  116. R. Lowen, “Compactness properties of the fuzzy real line,”Fuzzy Sets Syst.,13, 231–236 (1984).

    Article  Google Scholar 

  117. R. Lowen, “Fuzzy integers, fuzzy rationals and other subspaces of the fuzzy real line,”Fuzzy Sets Syst.,14, 231–236 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  118. R. Lowen, “The order aspect of the fuzzy real line,”Manuscr. Math.,39, 239–309 (1985).

    MathSciNet  Google Scholar 

  119. R. Lowen, “The relation between filter and net convergence in fuzzy topological spaces,”Fuzzy Math.,3, 41–52 (1985).

    MathSciNet  Google Scholar 

  120. R. Lowen, “Mathematics of fuzziness: some personal reflections,”Intp. Sci.,35, 17–27 (1985).

    MathSciNet  Google Scholar 

  121. R. Lowen, “On the existence of natural fuzzy topologies on spaces of probability measures,”Math. Nachr.,115, 33–57 (1984).

    MATH  MathSciNet  Google Scholar 

  122. R. Lowen,On the Existence of Natural Non-Topological Fuzzy Topological Spaces, Heldermann Verlag, Berlin (1985).

    Google Scholar 

  123. R. Lowen, “Concerning constants in fuzzy topology,”J. Math. Anal. Appl.,129, 256–268 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  124. R. Lowen and P. Wuyts, “Completeness, compactness and precompactness in fuzzy topological spaces. Part I,”J. Math. Anal. Appl.,90, 563–581 (1982).

    Article  MathSciNet  Google Scholar 

  125. R. Lowen and P. Wuyts, “Completeness, compactness and precompactness in fuzzy topological spaces. Part II,”J. Math. Anal. Appl.,91, 342–371 (1983).

    MathSciNet  Google Scholar 

  126. R. Lowen, P. Wuyts, and E. Lowen, “On reflectiveness and coreflectiveness of subcategories in FTS,”Math. Nachr.,141, 55–65 (1989).

    MathSciNet  Google Scholar 

  127. J. Lukasiewicz and A. Tarski, “Untersuchungen uber den Aussagenkalkül,”Comptes Rendus Seancea Soc. Scientes et Letteres Varsovie (III),23, 30–50 (1930).

    Google Scholar 

  128. M. Macho Stadler and M. A. de Parda Vicente, “Fuzzyt-net theory,”Fuzzy Sets Syst.,37, 225–235 (1998).

    Google Scholar 

  129. M. Macho Stadler and M. A. de Parda Vicente, “On N-convergence of fuzzy nets,”Fuzzy Sets Syst.,51, 203–217 (1992).

    Google Scholar 

  130. M. Macho Stadler and M. A. de Parda Vicente, “t-prefilter theory,”Fuzzy Sets Syst.,38, 115–124 (1990).

    Google Scholar 

  131. M. Macho Stadler and M. A. de Parda Vicente, “Fuzzy N-limit spaces,”Suppl. Rend. Mat. Palermo, No. 29, 521–528 (1992).

    Google Scholar 

  132. K. D. Magill, “A survey of semigroups of continuous selfmaps,”Semigroup Forum,11, 189–282 (1975–1976).

    MathSciNet  Google Scholar 

  133. S. Markin and A. Šostak, “Another approach to the concept of a fuzzy proximity,”Suppl. Rend. Mat. Palermo, No. 29, 530–551 (1992).

    Google Scholar 

  134. H. W. Martin, “A Stone-Čech ultrafuzzy compactization,”J. Math. Anal. Appl.,73, 453–456 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  135. H. W. Martin, “Weakly induced fuzzy topological spaces,”J. Math. Anal. Appl.,78, 634–639 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  136. A. S. Masshour, M. H. Ghanim, A. N. El-Wakeil, and N. N. Morsi, “On product fuzzy topological spaces,”Fuzzy Sets Syst.,30, 175–191 (1989); “Corrigendum,”31 131 (1989).

    Google Scholar 

  137. A. S. Masshour, M. H. Ghanim, A. N. El-Wakeil, and N. N. Morsi, “On productivity classes of fuzzy topology,”Fuzzy Sets Syst.,53, 193–202 (1993).

    Google Scholar 

  138. A. S. Masshour and N. N. Morsi, “On regularity axiom in fuzzy neighborhood spaces,”Fuzzy Sets Syst.,44, 265–271 (1991).

    Google Scholar 

  139. A. S. Masshour and N. N. Morsi, “Fuzzy metric neighborhood spaces,”Fuzzy Sets Syst.,45, 367–388 (1992).

    Google Scholar 

  140. A. S. Masshour, N. N. Morsi, and A. N. El-Wakeil, “Fuzzy neighborhood syntopogeneous structures,”Fuzzy Sets Syst.,43, 219–234 (1991).

    Google Scholar 

  141. P. P. Matutu,(L, M)-Fuzzy Topological Spaces (MSc Thesis), Rhodes Univ., Grahamstown, South Africa (1992).

    Google Scholar 

  142. Meng Guangwu, “Lowen's compactness in 699-1 topological spaces,”Fuzzy Sets Syst.,53, 319–334 (1993).

    Article  Google Scholar 

  143. K. Menger, “Statistical Metrics,”Proc. Natl. Acad. Sci. USA,28, 535–537 (1942).

    MATH  MathSciNet  Google Scholar 

  144. Min Kyung Chan, “Fuzzy limit spaces,”Fuzzy Sets Syst.,32, 343–357 (1989).

    Google Scholar 

  145. Mingsheng Ying, “A new approach to fuzzy topology. Part I,”Fuzzy Sets Syst.,39, 303–321 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  146. Mingsheng Ying, “A new approach to fuzzy topology. Part II,”Fuzzy Sets Syst.,47, 221–232 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  147. Mingsheng Ying, “A new approach to fuzzy topology. Part III,”Fuzzy Sets Syst.,55, 193–207 (1993).

    MATH  MathSciNet  Google Scholar 

  148. Mingsheng Ying, “Compactness in fuzzifying topologies,”Fuzzy Sets Syst.,55, 79–92 (1993).

    MATH  MathSciNet  Google Scholar 

  149. Mingsheng Ying, “Fuzzifying uniform spaces,”Fuzzy Sets Syst.,53, 93–104 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  150. N. N. Morsi, “A short note on fuzzy neighborhood spaces,”Fuzzy Sets Syst.,23, 393–397 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  151. N. N. Morsi, “Nearness concepts in fuzzy neighborhood spaces and their fuzzy proximity spaces,”Fuzzy Sets Syst.,31, 83–109 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  152. N. N. Morsi and S. E. Yehia, “Continuity of fuzzy multiplication in the N-Euclidean space,”Fuzzy Sets Syst.,46, 97–106 (1992).

    Article  MathSciNet  Google Scholar 

  153. S. A. Naimpally and B. D. Warrack,Proximity Spaces, Cambridge University Press, Cambridge (1970).

    Google Scholar 

  154. C. V. Negoita and D. A. Ralescu,Application of Fuzzy Sets to System Analysis, John Wiley & Sons, New York (1975).

    Google Scholar 

  155. G. Nöbling,Grundlagen der Analytischen Topologie, Springer Verlag, Berlin (1954).

    Google Scholar 

  156. M. A. de Prada Vicente, “Entornos de puntos fuzzy y continuidad,” In:Actas VIII Jornadas Luso-Espanolas Mathematicas. Vol. 1, Univ. Coimbra (1981), pp. 371–382.

  157. M. A. de Prada Vicente and M. Saralegui Aranguren, “Fuzzy filters,”J. Math. Anal. Appl.,129, 560–568 (1988).

    MathSciNet  Google Scholar 

  158. Pu Paoming and Liu Yingming, “Fuzzy topology. Part I. Neighborhood structure of a fuzzy point and Moore-Smith convergence,”J. Math. Anal. Appl.,76, 571–599 (1980).

    Article  MathSciNet  Google Scholar 

  159. Pu Paoming and Liu Yingming, “Fuzzy topology. Part I. Product and quotent spaces,”J. Math. Anal. Appl.,77, 20–37 (1980).

    Article  MathSciNet  Google Scholar 

  160. S. E. Rodabaugh, “The Hausdorff separation axiom for fuzzy topological spaces,”Topology Appl.,11, 314–334 (1980).

    Article  MathSciNet  Google Scholar 

  161. S. E. Rodabaugh, “The\(\mathbb{L}\)-fuzzy real line and its subspaces,” In:Fuzzy Sets and Possibility Theory (R. R. Yager, eds.), Pergamon Press, Oxford (1982), pp. 113–121.

    Google Scholar 

  162. S. E. Rodabaugh, “Connectivity and the\(\mathbb{L}\)-fuzzy unit interval,”Rocky Mount. J. Math.,12, 113–121 (1982).

    MATH  MathSciNet  Google Scholar 

  163. S. E. Rodabaugh, “The categorical accommodation of various notions to fuzzy topology,”Fuzzy Sets Syst.,9, 163–183 (1983).

    Article  Google Scholar 

  164. S. E. Rodabaugh, “Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines,”Fuzzy Sets Syst.,15, 285–310 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  165. S. E. Rodabaugh, “A theory of fuzzy uniformities with applications to the fuzzy real lines,”J. Math. Anal. Appl.,129, 37–70 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  166. S. E. Rodabaugh, “Lowen, para-Lowen and α-level functors and fuzzy topologies on the crisp real line,”J. Math. Anal. Appl.,131, 157–169 (1988).

    MATH  MathSciNet  Google Scholar 

  167. S. E. Rodabaugh, “Point-set lattice-theoretic topology,”Fuzzy Sets Syst.,40, 297–345 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  168. S. E. Rodabaugh, “Categorical framework for Stone representation theories,” In:Applications of Category Theory to Fuzzy Subsets (S. E. Rodabaugh, E. P. Klement, and U. Höhle, eds.), Kluwer Academic Publ. (1992), pp. 177–232.

  169. S. E. Rodabaugh, “Separation axioms and the fuzzy real lines,”Fuzzy Sets Syst.,11, 163–183 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  170. M. Sarkar, “On fuzzy topological spaces,”J. Math. Anal. Appl.,79, 384–394 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  171. M. Sarkar, “On\(\mathbb{L}\)-fuzzy topological spaces,”J. Math. Anal. Appl.,84, 431–442 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  172. B. Schweizer and A. Sclar, “Statistical metric spaces,”Pacific J. Math.,10, 313–334 (1960).

    MathSciNet  Google Scholar 

  173. B. Schweizer and A. Sclar,Probabilistic Metric Spaces, North Holland (1982).

  174. V. B. Shteinbuk and A. P. Šostak, “On semigroups of continuous transformations of fuzzily structured sets,”Semigroup Forum,43, 30–42 (1989).

    Google Scholar 

  175. V. B. Shteinbuk and A. P. Šostak, “On endomorphism semigroups of fuzzily structured sets,”Suppl. Kybernetica,28, 54–57 (1992).

    Google Scholar 

  176. A. P. Šostak, “On a fuzzy topological structure,”Suppl. Rend. Circ. Matem. Palermo, No. 11, 89–103 (1985).

    Google Scholar 

  177. A. P. Šostak, “On compactness and connectedness degrees of fuzzy sets in fuzzy topological spaces,” In:General Topology and Relat. Modern Analysis and Algebra. Proc. V-th Prague Topol. Symp., Prague, 1986, Heidermann Verlag, Berlin (1988), pp. 519–532.

    Google Scholar 

  178. A. P. Šostak, “On some modifications of fuzzy topologies,”Mat. Vesnik,41, 20–37 (1989).

    Google Scholar 

  179. A. P. Šostak, “On a category for fuzzy topology,”Zb. Radova Univ. u Nisu, Ser. Math.,2, 61–67 (1988).

    Google Scholar 

  180. A. P. Šostak, “Two decades of fuzzy topology: basic ideas, notions and results,”Russ. Math. Surveys,44, No. 6, 125–186 (1989).

    Google Scholar 

  181. A. P. Šostak, “On the neighborhood structure of a fuzzy topological space,”Zb. Radova Univ. u Nisu, Ser. Math.,4, 7–14 (1990).

    Google Scholar 

  182. A. P. Šostak, “On the convergence structure of a fuzzy topological space,” In:II Conference Balcanic Union for Fuzzy Systems and Artificial Intelligence, Trabzon, Turkey, Abstracts (1992), pp. 146–149.

  183. A. P. Šostak, “On complete regularity and E-regularity in fuzzy topology,”Mat. Vesnik,41, 189–203 (1989).

    Google Scholar 

  184. A. P. Šostak, “On a fuzzy proximity structure,” In:Proc. East Category Seminar, Predela, Bulgaria (1989), pp. 51–52.

  185. A. P. Šostak, “Fuzzy stratifiable spaces,”Math. Bohemica,2, No. 117, 169–184 (1992).

    Google Scholar 

  186. A. P. Šostak, “On fuzzy syntopogeneous structures,”Quaest. Math. (1994), in press.

  187. A. P. Šostak, “Introduction to the general theory of\(\mathbb{L}\)-fuzzy topological spaces,” In:Proc. Math. Seminar Partas Univ., Partas, Greece, Vol. 17 (1994), in press.

  188. A. P. Šostak, “A fuzzy modification of a linearly ordered space,”Topology Appl.,41, 581–604 (1983).

    Google Scholar 

  189. A. P. Šostak, “A fuzzy modification of the category of linearly ordered spaces,”Comm. Math. Univ. Carol.,26, 421–442 (1985).

    Google Scholar 

  190. R. Srivastava, S. N. Lal, and A. K. Srivastava, “Fuzzy Hausdorff topological spaces,”J. Math. Anal. Appl.,81, 497–506 (1981).

    MathSciNet  Google Scholar 

  191. R. Srivastava, S. N. Lal, and A. K. Srivastava, “Fuzzy T1-topological spaces,”J. Math. Anal. Appl.,102, 442–448 (1982).

    MathSciNet  Google Scholar 

  192. R. Srivastava and A. K. Srivastava, “On fuzzy Hausdorffness concepts,”Fuzzy Sets Syst.,17, 67–71 (1985).

    Google Scholar 

  193. L. N. Stout, “Topological space objects in a topos. II: ε-completeness and ε-completeness,”Manuscr. Math.,17, 1–14 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  194. Wang Guojun, “Topological molecular lattices,”Fuzzy Sets Syst.,47, 351–376 (1992).

    Google Scholar 

  195. Wang Guojun, “Some characteristic theorems of order homomorphism,”Sci. Bull,30, 723–728 (1985).

    Google Scholar 

  196. Wang Guojun, “Topological molecular lattices (II),”Kexue Tongbao,29, 19–29 (1984).

    Google Scholar 

  197. M. W. Warner, “Fuzzy topology with respect to continuous lattices,”Fuzzy Sets Syst.,35, 85–91 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  198. M. W. Warner, “Fuzzy points and membership,” In:Applications of Category to Fuzzy Subsets (S. E. Rodabaugh, E. P. Klement, and U. Höhle, eds.), Kluwer Acad. Publ. (1992), pp. 325–334.

  199. M. W. Warner, “Frame-fuzzy points and membership,”Fuzzy Sets Syst.,42, 335–344 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  200. R. H. Warren, “Neighborhood, bases and continuity in fuzzy topological spaces,”Rocky Mount. J. Math.,8, 459–470 (1978).

    MATH  MathSciNet  Google Scholar 

  201. R. H. Warren, “Fuzzy topologies characterized by neighborhood systems,”Rocky Mount. J. Math.,9, 761–764 (1979).

    MATH  MathSciNet  Google Scholar 

  202. R. H. Warren, “Convergence in fuzzy topology,”Rocky Mount. J. Math.,13, 31–36 (1983).

    MATH  MathSciNet  Google Scholar 

  203. A. Weil,Sur les Espaces a Structure Uniforme et sur la Topologie Generale, Paris (1938).

  204. M. D. Weiss, “Fixed points, separation and induced topologies for fuzzy sets,”J. Math. Anal. Appl.,50, 142–150 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  205. C. K. Wong, “Fuzzy topology: Product and quotient theorems,”J. Math. Anal. Appl.,45, 512–521 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  206. C. K. Wong, “Fuzzy points and local properties of fuzzy topology,”J. Math. Anal. Appl.,46, 316–328 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  207. Wu Cogxin and Wu Jianrong, “Fuzzy quasi-uniformities and fuzzy topological spaces,”Fuzzy Sets Syst.,46, 133–137 (1992).

    Google Scholar 

  208. P. Wuyts, “On determination of fuzzy topological spaces and fuzzy neighborhood spaces,”Fuzzy Sets Syts.,12, 71–85 (1984).

    MATH  MathSciNet  Google Scholar 

  209. P. Wuyts and R. Lowen, “On separation axioms in fuzzy topological spaces, fuzzy neighborhood spaces and fuzzy uniform spaces,”J. Math. Anal. Appl.,93, 27–41 (1983).

    Article  MathSciNet  Google Scholar 

  210. L. A. Zadeh, “Fuzzy sets,”Inf. Control.,8, 338–365 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  211. Zhang De-xue, “The stratified canonical fuzzy topology on I(L) is uniformizable,”Fuzzy Sets Syst.,54, 225–228 (1993).

    Article  Google Scholar 

  212. Zhao Dongsheng, “TheN-compactness in\(\mathbb{L}\)-fuzzy topological spaces,”J. Math. Anal. Appl.,108, 64–79 (1987).

    Google Scholar 

  213. Zhao Xiaodong, “Local property, normality and uniformity in fuzzy topological spaces,”J. Math. Anal. Appl.,127, 285–298 (1987).

    Article  MathSciNet  Google Scholar 

  214. Zhaojiang Chen, “The accumulation points in topological molecular lattices,”J. Nanjing Univ. Math. Biquart,5, No. 1, 42–49 (1988).

    MATH  Google Scholar 

  215. Zheng Chongyou and Shi Fugui, “Connectivity in fuzzy topological molecular lattices,”Fuzzy Sets Syst.,29, 363–370 (1989).

    Google Scholar 

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i ee Prilozheniya. Tematicheskiye Obzory. Vol. 14, Topologiya-2, 1994

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Šostak, A.P. Basic structures of fuzzy topology. J Math Sci 78, 662–701 (1996). https://doi.org/10.1007/BF02363065

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