On measures of compactness in fuzzy topological spaces

Lowen, E.; Löwen, R.

doi:10.1016/0022-247x(88)90209-0

Cited by 14 publications

(7 citation statements)

References 12 publications

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“…The inequality rc(b) ≤ λ rc(pr λ (b)) follows at once with the continuity of the pr λ , by Proposition 6.14. On the other hand, let We mention without proof that in the case of a = 1 X , (X, ∆) a fuzzy topological space, the compactness degree for * = T m is just the degree of compactness in E. Lowen and R. Lowen [11]. In this way, the compactness degrees here not only generalize the theory of compactness in FCS but also generalize the theory of compactness degrees in FTS, the category of fuzzy topological spaces.…”

Section: Corollary 615 (I) the Compactness Degree Of The Image Of Amentioning

confidence: 84%

“…Degrees of compactness and of relative compactness. In this section, we extend the theory of relative compact subsets established in [2,9] and repeat, sketching new proofs, the theory of compactness degrees developed in [10] (which extends the theory of compactness in fuzzy convergence spaces [6] and the theory of measures of compactness in [0, 1]-topological spaces [11]). Some additional results concerning compactness degrees are included.…”

Section: I) If ϕ Is Continuous Then Cl(ϕ ← (E)) ≥ Cl(e) (Ii) If ϕ Ismentioning

confidence: 98%

“…There are some papers dealing with such approaches. E. Lowen and R. Lowen [11] consider compactness degrees, and in [19], measures of separation in [0, 1]-topological spaces are investigated. In [17], Šostak developed a theory of compactness degrees and connectedness degrees in [0, 1]-fuzzy topological spaces, and developed, in [18], a theory of degrees of precompactness and completeness in the so-called Hutton fuzzy uniform spaces.…”

Section: Introductionmentioning

confidence: 99%

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Fuzzy properties in fuzzy convergence spaces

Jäger

2002

International Journal of Mathematics and Mathematical Sciences

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Based on the concept of limit of prefilters and residual implication, several notions in fuzzy topology are fuzzyfied in the sense that, for each notion, the degree to which it is fulfilled is considered. We establish therefore theories of degrees of compactness and relative compactness, of closedness, and of continuity. The resulting theory generalizes the corresponding "crisp" theory in the realm of fuzzy convergence spaces and fuzzy topology.2000 Mathematics Subject Classification: 54A20, 54A40, 54C05, 54D30. Introduction.In most papers and contributions to the theory of [0, 1]-topological spaces, the considered properties (like compactness) are viewed in a crisp way, that is, the properties either hold or fail. In [16], R. Lowen suggested that also the properties should be considered fuzzy, that is, one should be able to measure a degree to which a property holds. There are some papers dealing with such approaches. E. Lowen and R. Lowen , a theory of degrees of precompactness and completeness in the so-called Hutton fuzzy uniform spaces. These latter theories are related to the present work as they are explicitly based on a generalized inclusion (which is, however, not resulting from a residual implication).In this paper, we follow these ideas in a systematic way. Starting from the notion of limit of a prefilter as defined in [15], we consider a semigroup operation * on [0, 1] which is finitely distributive over arbitrary joins and, therefore, has a right adjoint →, that is, a residual implication operator. In this way, a natural way of obtaining truly fuzzy extensions of properties in fuzzy convergence spaces [12,13] is to replace subsethood, a ≤ b of two fuzzy sets a, b. Exploiting this idea leads to the theory considered in this paper. We extend some results of an earlier paper [10], where degrees of closedness and degrees of compactness were studied and a theory of degrees of continuity and of degrees of relative compactness is established. Note that stratified [0, 1]-topological spaces [5,14] as well as Choquet convergence spaces [3] are fuzzy convergence spaces [13], that is, our approach works also in this more special context.

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Section: Corollary 615 (I) the Compactness Degree Of The Image Of Amentioning

confidence: 84%

Section: I) If ϕ Is Continuous Then Cl(ϕ ← (E)) ≥ Cl(e) (Ii) If ϕ Ismentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Fuzzy properties in fuzzy convergence spaces

Jäger

2002

International Journal of Mathematics and Mathematical Sciences

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“…Chang's compactness has been greatly extended to the variable-basis case by Rodabaugh [11]. With the development of fuzzy topology, many researchers have successfully generalized the compactness theory of general topology to fuzzy setting (see [2,4,6,[8][9][10][13][14][15][16][17]). In 1985,Sostak [16] introduced a definition of the compact degree for a fuzzy set by means of the level I-topology.…”

Section: Introductionmentioning

confidence: 99%

Degrees of countable compactness and the Lindelöf property of L-fuzzy sets

Liang

2015

Journal of Intelligent &Amp; Fuzzy Systems

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In this paper, we introduce the notions of the degrees of countable compactness and the Lindelöf property in L-fuzzy topological spaces. Most of elementary results related to countable compactness and the Lindelöf property in general topology can be extended to L-fuzzy topological spaces. Moreover, many characterizations of the degrees of countable compactness and the Lindelöf property are presented.

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“…However, considering compactness can be a matter of degree, E. Lowen and R. Lowen [10] introduced the notion of compactness degrees of I-topological spaces. G. Jäger generalized it to fuzzy convergence spaces in [5].…”

Section: Introductionmentioning

confidence: 99%

Measures of compactness in L-fuzzy pretopological spaces

Shi

Liang

2014

Journal of Intelligent &Amp; Fuzzy Systems

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In this paper, we introduce the concepts of the degrees of compactness, countable compactness and Lindelöf property in L-fuzzy pretopological spaces by means of implication operator. These definitions do not rely on the structure of the basis lattice L and no distributivity in L is required. The notions of pre-compactness and semi-compactness in L-fuzzy topological spaces can be viewed as special cases of compactness in L-fuzzy pretopological spaces. Their properties are investigated. Further when L is completely distributive, their characterizations are presented.

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On measures of compactness in fuzzy topological spaces

Cited by 14 publications

References 12 publications

Fuzzy properties in fuzzy convergence spaces

Fuzzy properties in fuzzy convergence spaces

Degrees of countable compactness and the Lindelöf property of L-fuzzy sets

Measures of compactness in L-fuzzy pretopological spaces

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