Some Remarks on Fuzzy <i>sb</i>-Metric Spaces

Fuzzy strong <i>b</i>-metrics called here by fuzzy sb-metrics, were introduced recently as a fuzzy version of strong <i>b</i>-metrics. It was shown that open balls in fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s&l...

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Main Authors: Tarkan Öner, Alexander Šostak
Format: Article
Language:English
Published: MDPI AG 2020-11-01
Series:Mathematics
Subjects:
Online Access:https://www.mdpi.com/2227-7390/8/12/2123
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author Tarkan Öner
Alexander Šostak
author_facet Tarkan Öner
Alexander Šostak
author_sort Tarkan Öner
collection DOAJ
description Fuzzy strong <i>b</i>-metrics called here by fuzzy sb-metrics, were introduced recently as a fuzzy version of strong <i>b</i>-metrics. It was shown that open balls in fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metric spaces are open in the induced topology (as different from the case of fuzzy <i>b</i>-metric spaces) and thanks to this fact fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metrics have many useful properties common with fuzzy metric spaces which generally may fail to be in the case of fuzzy <i>b</i>-metric spaces. In the present paper, we go further in the research of fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metric spaces. It is shown that the class of fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metric spaces lies strictly between the classes of fuzzy metric and fuzzy <i>b</i>-metric spaces. We prove that the topology induced by a fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metric is metrizable. A characterization of completeness in terms of diameter zero sets in these structures is given. We investigate products and coproducts in the naturally defined category of fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metric spaces.
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spelling doaj.art-8bcb57d407324578a2eadde726afb4ff2023-11-20T22:33:05ZengMDPI AGMathematics2227-73902020-11-01812212310.3390/math8122123Some Remarks on Fuzzy <i>sb</i>-Metric SpacesTarkan Öner0Alexander Šostak1Department of Mathematics, Muğla Sıtkı Koçman University, Muğla 48000, TurkeyInstitute of Mathematics and CS and Department of Mathematics, University of Latvia, LV-1586 Riga, LatviaFuzzy strong <i>b</i>-metrics called here by fuzzy sb-metrics, were introduced recently as a fuzzy version of strong <i>b</i>-metrics. It was shown that open balls in fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metric spaces are open in the induced topology (as different from the case of fuzzy <i>b</i>-metric spaces) and thanks to this fact fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metrics have many useful properties common with fuzzy metric spaces which generally may fail to be in the case of fuzzy <i>b</i>-metric spaces. In the present paper, we go further in the research of fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metric spaces. It is shown that the class of fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metric spaces lies strictly between the classes of fuzzy metric and fuzzy <i>b</i>-metric spaces. We prove that the topology induced by a fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metric is metrizable. A characterization of completeness in terms of diameter zero sets in these structures is given. We investigate products and coproducts in the naturally defined category of fuzzy <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mi>b</mi></mrow></semantics></math></inline-formula>-metric spaces.https://www.mdpi.com/2227-7390/8/12/2123fuzzy metricfuzzy <i>sb</i>-metricfuzzy <i>b</i>-metric
spellingShingle Tarkan Öner
Alexander Šostak
Some Remarks on Fuzzy <i>sb</i>-Metric Spaces
Mathematics
fuzzy metric
fuzzy <i>sb</i>-metric
fuzzy <i>b</i>-metric
title Some Remarks on Fuzzy <i>sb</i>-Metric Spaces
title_full Some Remarks on Fuzzy <i>sb</i>-Metric Spaces
title_fullStr Some Remarks on Fuzzy <i>sb</i>-Metric Spaces
title_full_unstemmed Some Remarks on Fuzzy <i>sb</i>-Metric Spaces
title_short Some Remarks on Fuzzy <i>sb</i>-Metric Spaces
title_sort some remarks on fuzzy i sb i metric spaces
topic fuzzy metric
fuzzy <i>sb</i>-metric
fuzzy <i>b</i>-metric
url https://www.mdpi.com/2227-7390/8/12/2123
work_keys_str_mv AT tarkanoner someremarksonfuzzyisbimetricspaces
AT alexandersostak someremarksonfuzzyisbimetricspaces