Fuzzy pseudometric spaces vs fuzzifying structures

Mardones-Pérez, I.; Vicente, María Ángeles de Prada

doi:10.1016/j.fss.2014.06.003

Cited by 18 publications

(11 citation statements)

References 15 publications

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“…Definition 7.1 ([40], cf. [39,30]). A fuzzifying uniformity on a nonempty set X is a mapping U : D(X) → I such that:…”

Section: Definition 64 ([25]mentioning

confidence: 99%

“…On the other hand, in [12] the authors associated to each fuzzy metric space a Hutton quasi-uniformity (see Proposition 5.8) and this association factorizes, via a Katsaras' functor, the association of uniform spaces to fuzzy metric spaces. Furthermore, Yue and Shi [40] constructed a fuzzifying uniformity by means of a fuzzy pseudometric (see also [30]). Recently, Yue and Fang [41] have constructed an I-uniformity in a fuzzy metric space in the sense of Kramosil and Michalek.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, the last section is devoted to the study of the relationship between fuzzy metrics and fuzzifying topologies and uniformities. In particular, we develop another method of constructing a probabilistic uniformity in a fuzzy metric space by using the fuzzifying uniformity constructed in this context as in [30,40]. We also clarify its relationship with the other probabilistic uniformities induced by a fuzzy metric.…”

Section: Introductionmentioning

confidence: 99%

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On fuzzy uniformities induced by a fuzzy metric space

García

López

Romaguera

2018

Fuzzy Sets and Systems

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Different types of fuzzy uniformities have been introduced in the literature standing out the notions due to Hutton, Höhle and Lowen. The main purpose of this paper is to study several methods to endow a fuzzy metric space (X, M, *), in the sense of George and Veeramani, with a probabilistic uniformity and with a Hutton [0, 1](-quasi)-uniformity. We will show the functorial behaviour of these constructions as well as its relation with respect to Lowen's functors and Katsaras's functors, which establish a relationship between the categories of probabilistic uniformities and Hutton [0, 1](-quasi)-uniformities with the category of classical uniformities respectively. Furthermore, we also study the fuzzy topologies associated with these fuzzy uniformities.

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“…Definition 7.1 ([40], cf. [39,30]). A fuzzifying uniformity on a nonempty set X is a mapping U : D(X) → I such that:…”

Section: Definition 64 ([25]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On fuzzy uniformities induced by a fuzzy metric space

García

López

Romaguera

2018

Fuzzy Sets and Systems

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“…In particular, it has been proved that the class of quasi-metrizable topological spaces and the class of fuzzy quasi-metrizable topological spaces agree with each other. These notions have been widely used in various papers devoted to fuzzy topology [7,10,13,15,16,25,28]. Meanwhile, much research has focused on investigating the interplay between fuzzy measures, domain theory and fuzzy metric spaces [14,17,18,19,20,24,26,27].…”

Section: Introductionmentioning

confidence: 99%

Formal balls in fuzzy quasi-metric spaces

Gao¹,

Li²,

Guo³

et al. 2017

J. Nonlinear Sci. Appl.

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The notions of Yoneda completeness and Smyth completeness on fuzzy quasi-metric spaces are introduced and their relationship with other types of completeness including sequentially Yoneda completeness and bicompleteness are investigated. Then we use the standard Yoneda completeness to characterize the order-theoretical properties of the poset (BX, M ) of formal balls in a fuzzy quasi-metric space (X, M, ∧). The results show that if (BX, M ) is a dcpo, then (X, M, ∧) is standard complete and conversely, (BX, M ) forms a dcpo provided that (X, M, ∧) is standard Yoneda complete. Particularly, in a fuzzy metric space, we clarify three types of completeness which can be characterized by the directed completeness of the related poset of formal balls.

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“…Recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. In particular in the papers [37], [22], [26] fuzzy pseudometrics m : X × X × (0, ∞) were applied to induce fuzzifying topologies T : 2 X → [0, 1]. However, as far as we know, there was still no research in the field of "fullbodied" fuzzy topologies T : [0,1] X → [0, 1] induced by fuzzy metrics.…”

Section: Introductionmentioning

confidence: 99%

A construction of a fuzzy topology from a strong fuzzy metric

2016

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After the inception of the concept of a fuzzy metric by I. Kramosil and J. Michalek, and especially after its revision by A. George and G. Veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. In most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. Recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. In particular, in the paper (J. J. Miñana, A. Sostak, Fuzzifying topology induced by a strong fuzzy metric, Fuzzy Sets and Systems 300 (2016), 24-39) a fuzzifying topology T : 2 X → [0, 1] induced by a fuzzy metric m : X × X × [0, ∞) was constructed. In this paper we extend this construction to get the fuzzy topology T : [0,1] X → [0, 1] and study some properties of this fuzzy topology.2010 MSC: 54A40.

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Fuzzy pseudometric spaces vs fuzzifying structures

Cited by 18 publications

References 15 publications

On fuzzy uniformities induced by a fuzzy metric space

On fuzzy uniformities induced by a fuzzy metric space

Formal balls in fuzzy quasi-metric spaces

A construction of a fuzzy topology from a strong fuzzy metric

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