Separation axioms, subspaces and sums in fuzzy topology

Ghanim, M. H.; Kerre, Etienne; Mashhour, A. S.

doi:10.1016/0022-247x(84)90212-9

Cited by 66 publications

(36 citation statements)

References 9 publications

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“…x p is called the support of p and p(x p )=t the value of p. However, we shall agree that a fuzzy crisp point q in X is a fuzzy set in X given by q(x) =1 for x=x q and q(x)= 0 for x ≠ x q. It is clear that p is a fuzzy singleton (as introduced by Goguen [5] and used by Ghanim, Kerre and Mashhour [4]) if and only if p is either a fuzzy point or a fuzzy crisp point. We shall also follow [4] for the definition of…”

Section: Introductionmentioning

confidence: 91%

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The Number of Fuzzy Clopen Sets in Fuzzy Topological Spaces

Fora¹

2017

JMSA

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We show the number of fuzzy clopen sets in an arbitrary fuzzy topological space can be any natural number greater than 1 if it is finite. We give an upper bound for this number. We shall also prove that the number of all crisp fuzzy clopen sets in an arbitrary fuzzy topological space is a power of 2 if it is finite. Keywords: clopen, enumerating, finite set, fuzzy clopen*This research has taken place while the author has a sabbatical leave from Yarmouk University.

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Section: Introductionmentioning

confidence: 91%

“…It is clear that p is a fuzzy singleton (as introduced by Goguen [5] and used by Ghanim, Kerre and Mashhour [4]) if and only if p is either a fuzzy point or a fuzzy crisp point. We shall also follow [4] for the definition of…”

Section: Introductionmentioning

confidence: 99%

The Number of Fuzzy Clopen Sets in Fuzzy Topological Spaces

Fora¹

2017

JMSA

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“…In [10] all different separation axioms have been discussed in detail. In [4] a characterization of normality in fuzzy topology has been given as well as a full study of the normality of a fuzzy Sierpinski space [11].…”

Section: On the Separation Axioms In Chang Fuzzy Topological Spacesmentioning

confidence: 99%

An Overview of the Fuzzy Axiomatic Systems and Characterizations Proposed at Ghent University

Kerre

D'eer

Gasse

2016

Axioms

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During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research unit of Ghent University several axiomatic systems and characterizations have been introduced. In this paper we highlight some of them. The main purpose of this paper consists of an invitation to continue research on these first attempts to axiomatize important concepts and systems in fuzzy set theory. Currently, these attempts are spread over many journals; with this paper they are now collected in a neat overview. In the literature, many axiom systems have been introduced, but as far as we know the axiomatic system of Huntington concerning a Boolean algebra has been the only one where the axioms have been proven independent. Another line of further research could be with respect to the simplification of these systems, in discovering redundancies between the axioms. Keywords: fuzzy set; fuzzy preference structure; fuzzy topology OverviewWe start with the characterization of a Chang fuzzy topology by means of a preassigned operation such as an interior operator. In Section 3 we dwell upon the separation axioms in Chang fuzzy topological spaces and refer to dependencies between the different fuzzy separation axioms. In Section 4 a fuzzy extension of the well-known Armstrong axioms in a fuzzy relational database is given. A lot of work has been done on the characterization of a fuzzy preference structure. Some results are repeated in Section 5. In the next section a characterization is given of lattices that are needed to establish the equivalence between Goguen's L-fuzzy sets and Gentilhomme's L-flou sets, leading to the introduction of the kite-tail lattices. We also dwell upon the modifications of the condition that sometimes appear in the literature. Section 7 summarizes the work on the axiomatization of the ordering of fuzzy quantities, in particular fuzzy numbers. In the next section some axioms are introduced for a defuzzification technique in order to transform a fuzzy set into a single element of the underlying universe. The concept of a fuzzy implication is important from a theoretical as well as a practical point of view. In Section 9 we describe the extension of Smets-Magrez axioms for a fuzzy implication. Finally Section 10 treats the axiomatization of a triangle algebra. On the Characterization of a Chang Fuzzy Topology by Means of Preassigned OperationsPerhaps general topology has been the first mathematical structure that has been fuzzified. Already in 1968, three years after the publication of Zadeh's seminal paper "Fuzzy Sets" [1], Chang's paper "Fuzzy Topological Spaces" [2] appeared in JMAA. A fuzzy topology on a set X was defined as a class T of fuzzy sets satisfying the straightforward fuzzifications of the classical axioms for a fuzzy topology:

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“…Some of these differ in a nonessential way. Of the works where this problem is considered in the context L = I, one can mention papers by R.H. Warren [200,201] (his approach is based on the use of fuzzy neighborhoods of usual points) and papers [206,158,32,48,88,89,49] (all these authors work with fuzzy type structures of fuzzy points and their approaches differ only in the interpretation of the corresponding notions). A comparative analysis of some of the approaches of the last group was worked out by E.E.…”

Section: Z851mentioning

confidence: 99%

“…Now the direct sum can be defined as the pair (X, r) (cf [48]). Obviously, 7-can be characterized also as the strongest L-fuzzy topology on X, for which all natural embeddings ix: Xx -+ X are continuous (i.e., 7-is the final L-fuzzy topology for the family of all embeddings) and, therefore, (X, 7-)is indeed the coproduct of the given family of L-fuzzy spaces in eFT(L).…”

Section: Jej J~j In the Case Where L Is Equipped With An Involution mentioning

confidence: 99%

Basic structures of fuzzy topology

Šostak¹

1996

J Math Sci

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Separation axioms, subspaces and sums in fuzzy topology

Cited by 66 publications

References 9 publications

The Number of Fuzzy Clopen Sets in Fuzzy Topological Spaces

The Number of Fuzzy Clopen Sets in Fuzzy Topological Spaces

An Overview of the Fuzzy Axiomatic Systems and Characterizations Proposed at Ghent University

Basic structures of fuzzy topology

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