On some results of analysis for fuzzy metric spaces

George, Aravind; Veeramani, P.

doi:10.1016/s0165-0114(96)00207-2

Cited by 327 publications

(248 citation statements)

References 6 publications

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“…(X, M, * ) is said to be complete if every Cauchy sequence converges with respect to τ M (see e.g. [5]). Remark 1.…”

Section: Definition 1 ([11 Definition 7])mentioning

confidence: 99%

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A domain-theoretic approach to fuzzy metric spaces

Ricarte

Romaguera

2014

Topology and its Applications

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We introduce a partial order M on the set BX of formal balls of a fuzzy metric space (X, M, ∧) in the sense of Kramosil and Michalek, and discuss some of its properties. We also characterize when the poset (BX, M ) is a continuous domain by means of a new notion of fuzzy metric completeness introduced here. The well-known theorem of Edalat and Heckmann that a metric space is complete if and only if its poset of formal balls is a continuous domain, is deduced from our characterization.

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“…(X, M, * ) is said to be complete if every Cauchy sequence converges with respect to τ M (see e.g. [5]). Remark 1.…”

Section: Definition 1 ([11 Definition 7])mentioning

confidence: 99%

“…is a (complete) fuzzy metric space called the standard fuzzy metric space of (X, d) (compare [4,5]). Moreover, the topology τ M d coincides with the topology induced by d.…”

Section: Definition 1 ([11 Definition 7])mentioning

confidence: 99%

A domain-theoretic approach to fuzzy metric spaces

Ricarte

Romaguera

2014

Topology and its Applications

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“…If, in addition, (M, * ) satisfies the symmetry condition (KM5), then (X, M, * ) is a fuzzy metric space in the sense of George and Veeramani ( [4,5]), and we will refer to it as a GV-fuzzy metric space. Obviously, each GV-fuzzy (quasi-)metric space (X, M, * ) can be considered as a fuzzy (quasi-)metric space, in the sense of Definition 1, by defining M (x, y, 0) = 0 for all x, y ∈ X.…”

Section: Definition 2 ([19])mentioning

confidence: 99%

Weightable quasi-uniformities

Künzi

Romaguera

2012

Acta Math Hung

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Extending the well-known result that every fuzzy metric space, in the sense of Kramosil and Michalek, has a completion which is unique up to isometry, we show that every KM-fuzzy quasi-metric space has a bicompletion which is unique up to isometry, and deduce that for each KM-fuzzy quasi-metric space, the completion of its induced fuzzy metric space coincides with the fuzzy metric space induced by its bicompletion.

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“…George and Veeramani, [1,3], introduced and studied a notion of fuzzy metric with the help of continuous t-norms. If M is a fuzzy metric on a (non-empty) set X, a topology τ M is deduced from M .…”

Section: Introductionmentioning

confidence: 99%