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Characterization Of L-Topologies By L-Valued Neighborhoods

  • U. Höhle
Part of the The Handbooks of Fuzzy Sets Series book series (FSHS, volume 3)

Abstract

It is well known that L-topologies can be characterized by L-neighborhood systems (cf. Subsection 6.1 in [14]). The aim of this paper is to give a characterization of a subclass of L-topologies by crisp systems of L-valued neighborhoods. This subclass consists of stratified and transitive L-topologies and covers simultaneously probabilistic L-topologies and [0, l]-topologies determined by fuzzy neighborhood spaces. We present this characterization depending on the structure of the underlying lattice L. In the case of probabilistic L-topologies it is remarkable to see that the structure of complete MV-algebras (cf. [2, 12]) is sufficient, while in all other case the complete distributivity of the underlying lattice L seems to be essential. Further, if L is given by the real unit interval [0, 1], then the Booleanization of [0, l]-topologies corresponding to fuzzy neighborhood spaces exists. Hence fuzzy neighborhood spaces can be chararcterized by two different types of many valued neighborhoods — namely by Booelan valued neighborhoods or by [0, l]-valued neighborhoods as the name “fuzzy neighborhood space” suggests (cf. Remark 3.17, Proposition 5.1). Moreover, [0, l]-fuzzifying topologies and [0, l]-topologies of fuzzy neighborhood spaces are equivalent concepts. Finally, we underline the interesting fact that a special class of stratified and transitive, [0, l]-topological spaces is induced by Menger spaces which form an important subclass of probabilistic metric spaces (cf. Example 5.6).

Keywords

Distributive Lattice Complete Distributivity Underlying Lattice Equivalent Concept Fuzzy Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • U. Höhle
    • 1
  1. 1.Fachbereich 7 Mathematik BergischeUniversiät Wuppertal GaußstraßeWuppertalGermany

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