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Open AccessArticle

A Characterization of Strong Completeness in Fuzzy Metric Spaces

1
Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/ Paranimf, 1, 46730 Grao de Gandia, Spain
2
Departament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Carretera de Valldemossa km. 7.5, 07122 Palma, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(6), 861; https://doi.org/10.3390/math8060861
Received: 29 April 2020 / Revised: 19 May 2020 / Accepted: 21 May 2020 / Published: 26 May 2020
Here, we deal with the concept of fuzzy metric space ( X , M , ) , due to George and Veeramani. Based on the fuzzy diameter for a subset of X , we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory. View Full-Text
Keywords: fuzzy metric; Cauchy sequence; (strong) convergence; completeness; fuzzy diameter fuzzy metric; Cauchy sequence; (strong) convergence; completeness; fuzzy diameter
MDPI and ACS Style

Gregori, V.; Miñana, J.-J.; Roig, B.; Sapena, A. A Characterization of Strong Completeness in Fuzzy Metric Spaces. Mathematics 2020, 8, 861. https://doi.org/10.3390/math8060861

AMA Style

Gregori V, Miñana J-J, Roig B, Sapena A. A Characterization of Strong Completeness in Fuzzy Metric Spaces. Mathematics. 2020; 8(6):861. https://doi.org/10.3390/math8060861

Chicago/Turabian Style

Gregori, Valentín; Miñana, Juan-José; Roig, Bernardino; Sapena, Almanzor. 2020. "A Characterization of Strong Completeness in Fuzzy Metric Spaces" Mathematics 8, no. 6: 861. https://doi.org/10.3390/math8060861

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