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Yue, Yueli; Shi, Fu-Gui

Generalized quasi-proximities. (English) Zbl 1116.54005

Fuzzy Sets Syst. 158, No. 4, 386-398 (2007).
The authors study relationships between different notions of two important topological concepts: uniformity and proximity in fuzzy topological spaces. They prove that \(s\)-fuzzy remote neighborhood spaces are isomorphic to topological fuzzy remote neighborhood spaces. Moreover the same category of spaces is found to be isomorphic to \(s\)-fuzzy quasi proximity spaces and can be embedded as a bireflective full subcategory of fuzzy quasi proximity spaces and these in turn, can be embedded in fuzzy pointwise quasi uniform spaces as a bireflective full subcategory. The paper is very theoretical and there are no applications considered or even mentioned in it.
Reviewer: A. Šwierniak (Gliwice)

Cited in 1 Review
Cited in 2 Documents

MSC:

54A40 Fuzzy topology
54E05 Proximity structures and generalizations
54E15 Uniform structures and generalizations
54A05 Topological spaces and generalizations (closure spaces, etc.)
54E55 Bitopologies

Keywords:

L-fuzzy co-topology; fuzzy remote neighborhood system; generalized pointwise \(S\)-quasi-uniformity; generalized \(S\)-quasi-proximity; generalized quasi-proximity
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References:

[1] Adámek, J.; Herrlich, H.; Strecker, G. E., Abstract and Concrete Categories (1990), Wiley: Wiley New York · Zbl 0695.18001
[2] Artico, G.; Moresco, R., Fuzzy proximities and totally bounded fuzzy uniformities, J. Math. Anal. Appl., 99, 320-337 (1984) · Zbl 0558.54002
[3] Dzhajanbajev, D.; Šostak, A. P., On a fuzzy uniform structure, Acta et Comm. Univ. Tartu, 940, 31-36 (1992) · Zbl 0809.54008
[4] Ghanim, M. H.; Tantawy, O. A.; Selim, F. M., On \(S\)-fuzzy quasi-proximity spaces, Fuzzy Sets and Systems, 109, 285-290 (2000) · Zbl 0949.54009
[5] Gierz, G., A Compendium of Continuous Lattices (1980), Springer: Springer Berlin · Zbl 0452.06001
[6] Gutierrez Garcia, J.; de Prada Vicente, M. A.; Šostak, A. P., A unified approach to the concept of fuzzy \(L\)-uniform space, (Rodabaugh, S. E.; Klement, E. P., Topological and Algebraic Structures in Fuzzy Sets, Trends in Logic, vol. 20 (2003), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, Dordrecht, London), 81-114, (Chapter 3) · Zbl 1061.54006
[7] Höhle, U., Probabilistic metrization of fuzzy topologies, Fuzzy Sets and Systems, 8, 63-69 (1982) · Zbl 0494.54006
[8] Höhle, U.; Šostak, A. P., Axiomatic foundations of fixed-basis fuzzy topology, (Höhle, U.; Rodabaugh, S. E., Mathematics of Fuzzy Sets-Logic, Topology and Measure Theory (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, Dordrecht, London), 123-272 · Zbl 0977.54006
[9] Hutton, B., Uniformities on fuzzy topological spaces, J. Math. Anal. Appl., 58, 557-571 (1977) · Zbl 0358.54008
[10] Katsaras, A. K., Fuzzy proximities and fuzzy completely regular spaces, J. Anal. St. Univ. Jasi., 26, 31-41 (1980) · Zbl 0443.54007
[11] T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985.; T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985. · Zbl 0589.54013
[12] Liu, W. J., Fuzzy proximity spaces redefined, Fuzzy Sets and Systems, 30, 37-45 (1989) · Zbl 0672.54006
[13] Rodabaugh, S. E., A theory of fuzzy uniformities with applications to the fuzzy real lines, J. Math. Anal. Appl., 129, 37-70 (1988) · Zbl 0637.54007
[14] S.E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets-Logic, Topology and Measure Theory, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 91-116 (Chapter 2).; S.E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets-Logic, Topology and Measure Theory, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 91-116 (Chapter 2). · Zbl 0974.03047
[15] S. E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, in: Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets-Logic, Topology and Measure Theory, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 273-388 (Chapter 4).; S. E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, in: Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets-Logic, Topology and Measure Theory, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 273-388 (Chapter 4). · Zbl 0968.54003
[16] S.E. Rodabaugh, Axiomatic foundations for uniform operator quasi-uniformities, in: S.E. Rodabaugh, E.P. Kelment (Eds.), Topological and Algebraic Structures in Fuzzy Sets, Trends in Logic, vol. 20, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003, pp. 199-233 (Chapter 7).; S.E. Rodabaugh, Axiomatic foundations for uniform operator quasi-uniformities, in: S.E. Rodabaugh, E.P. Kelment (Eds.), Topological and Algebraic Structures in Fuzzy Sets, Trends in Logic, vol. 20, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003, pp. 199-233 (Chapter 7). · Zbl 1041.54012
[17] Shi, F. G., Pointwise uniformities in fuzzy set theory, Fuzzy Sets and Systems, 98, 141-146 (1998) · Zbl 0941.54009
[18] Shi, F. G., \(L\)-proximity and totally bounded pointwise quasi-uniformity, Fuzzy Sets and Systems, 133, 321-331 (2003) · Zbl 1041.54013
[19] Shi, F. G., The category of pointwise \(S\)-proximity spaces, Fuzzy Sets and Systems, 152, 349-372 (2005) · Zbl 1081.54008
[20] Šostak, A. P., On a fuzzy topological structure, Rend Circ. Mat. Palermo (Suppl. Ser. II), 11, 89-103 (1985) · Zbl 0638.54007
[21] Wang, G., Theory of topological molecular lattices, Fuzzy Sets and Systems, 47, 351-376 (1992) · Zbl 0783.54032
[22] Yue, Y.; Fang, J., Categories isomorphic to the Kubiak-Šostak extension of TML, Fuzzy Sets and Systems, 157, 832-842 (2006) · Zbl 1099.54009
[23] Yue, Y.; Fang, J., Extension of Shi’s quasi-uniformities in a Kubiak-Šostak sense, Fuzzy Sets and Systems, 157, 1956-1969 (2006) · Zbl 1111.54007
[24] Zhang, D., A comparison of various uniformities in fuzzy topology, Fuzzy Sets and Systems, 140, 399-422 (2003) · Zbl 1047.54008
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