Continuous triangular norm based fuzzy topology

  • Dexue ZhangEmail author
  • Gao Zhang


For each continuous t-norm &, a class of fuzzy topological spaces, called &-topological spaces, is introduced. The motivation stems from the idea that to each many-valued logic there may correspond a theory of many-valued topology, in particular, each continuous t-norm may lead to a theory of fuzzy topology. It is shown that for each continuous t-norm &, the subcategory consisting of &-topological spaces is simultaneously reflective and coreflective in the category of fuzzy topological spaces, hence gives rise to an autonomous theory of fuzzy topology. Topologizing a fuzzy pre-ordered set with the fuzzy Scott topology yields a functor from the category of fuzzy pre-ordered sets and maps that preserve suprema of flat ideals to the category of &-topological spaces. It is proved that this functor is a full one if and only if the t-norm & is Archimedean.


Continuous triangular norm Fuzzy topology Fuzzy pre-order Fuzzy Scott topology 

Mathematics Subject Classification

54A40 18B30 06F30 03B52 


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We thank sincerely Dr. Hongliang Lai for stimulating discussion during the preparation of this paper.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsSichuan UniversityChengduChina

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