Metrization of soft metric spaces and its application to fixed point theory

Gültekin Soylu; Müge Çerçi; Gültekin Soylu; Müge Çerçi

doi:10.3934/math.2024336

AIMS Mathematics

2024, Volume 9, Issue 3: 6904-6915. doi: 10.3934/math.2024336

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Research article

Metrization of soft metric spaces and its application to fixed point theory

Gültekin Soylu ^{1
,
,},
Müge Çerçi ²

1.
Department of Mathematics, Akdeniz University, Antalya 07058, Turkey
2.
Department of Mathematics, Haliç University, İstanbul, Turkey

Received: 21 August 2023 Revised: 18 October 2023 Accepted: 25 October 2023 Published: 19 February 2024
MSC : 47H10, 54A05, 54E35

Soft set theory has attracted many researchers from several different branches. Sound theoretical improvements are accompanied with successful applications to practical solutions of daily life problems. However, some of the attempts of generalizing crisp concepts into soft settings end up with completely equivalent structures. This paper deals with such a case. The paper mainly presents the metrizability of the soft topology induced by a soft metric. The soft topology induced by a soft metric is known to be homeomorphic to a classical topology. In this work, it is shown that this classical topology is metrizable. Moreover, the explicit construction of an ordinary metric that induces the classical topology is given. On the other hand, it is also shown that soft metrics are actually cone metrics. Cone metrics are already proven to be an unsuccessful attempt of generalizing metrics. These results clarify that most, if not all, properties of soft metric spaces could be directly imported from the related classical theory. The paper concludes with an application of the findings, i.e., a new soft fixed point theorem is stated and proven with the help of the obtained homemorphism.
- soft metric space,
- soft topology,
- topology,
- metrizability,
- fixed point
Citation: Gültekin Soylu, Müge Çerçi. Metrization of soft metric spaces and its application to fixed point theory[J]. AIMS Mathematics, 2024, 9(3): 6904-6915. doi: 10.3934/math.2024336

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Abstract

Soft set theory has attracted many researchers from several different branches. Sound theoretical improvements are accompanied with successful applications to practical solutions of daily life problems. However, some of the attempts of generalizing crisp concepts into soft settings end up with completely equivalent structures. This paper deals with such a case. The paper mainly presents the metrizability of the soft topology induced by a soft metric. The soft topology induced by a soft metric is known to be homeomorphic to a classical topology. In this work, it is shown that this classical topology is metrizable. Moreover, the explicit construction of an ordinary metric that induces the classical topology is given. On the other hand, it is also shown that soft metrics are actually cone metrics. Cone metrics are already proven to be an unsuccessful attempt of generalizing metrics. These results clarify that most, if not all, properties of soft metric spaces could be directly imported from the related classical theory. The paper concludes with an application of the findings, i.e., a new soft fixed point theorem is stated and proven with the help of the obtained homemorphism.

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