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Extensions Of Uniform Space Notions

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Mathematics of Fuzzy Sets

Part of the book series: The Handbooks of Fuzzy Sets Series ((FSHS,volume 3))

Abstract

A large part of mathematics is based on the notion of a set and on binary logic. Statements are either true or false and an element either belongs to a set or not. In order to accommodate the idea of a sliding transition between the two states: true and false, and to generalise the concept of a subset of a given set, Zadeh introduced the notion of a fuzzy subset in a now-famous paper: [52]. For the record, let us recall that if X is a set and A is a subset of X then the characteristic function, denoted 1 A , is defined by Thus 1 A ∈ 2X. In [52], an element μ ∈ I X, where I denotes the closed unit interval, was called a fuzzy set in X, with μ(x) being interpreted as the degree to which x belongs to the fuzzy set μ. Since the elements μ ∈ I X are generalisations of subsets of X, it is more accurate to refer to them as fuzzy subsets of X and we shall adopt this terminology here.

*

This work was completed during a visit of J. Gutiérrez García to Rhodes University in August 1997. The authors would like thank Prof. W. Kotzé for making the arrangements for the visit and the Foundation for Research Development of South Africa for their generous support.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Rhodes University Grahamstown, 6140, South Africa

    M. H. Burton

  2. Matematika Saila Euskal, Herriko Unibertsitatea, E-48080, Bilbo, Spain

    J. Gutiérrez García

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  1. M. H. Burton
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  2. J. Gutiérrez García
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Editor information

Editors and Affiliations

  1. Fachbereich Mathematik, Bergische Universität, Wuppertal, Germany

    Ulrich Höhle

  2. Department of Mathematics and Statistics, Youngstown State University, Youngstown, Ohio, USA

    Stephen Ernest Rodabaugh

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Burton, M.H., Gutiérrez García, J. (1999). Extensions Of Uniform Space Notions. In: Höhle, U., Rodabaugh, S.E. (eds) Mathematics of Fuzzy Sets. The Handbooks of Fuzzy Sets Series, vol 3. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5079-2_11

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  • DOI: https://doi.org/10.1007/978-1-4615-5079-2_11

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7310-0

  • Online ISBN: 978-1-4615-5079-2

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