Fuzzy topology with $  \mathcal{I} $-convergence

Abdelhak Razouki; Omar El Ogri; Jaouad EL-Mekkaoui; Zamir Ahmad Ansari; Naglaa F. Soliman; Abeer D. Algarni; Abdelhak Razouki; Omar El Ogri; Jaouad EL-Mekkaoui; Zamir Ahmad Ansari; Naglaa F. Soliman; Abeer D. Algarni

doi:10.3934/math.2025852

AIMS Mathematics

2025, Volume 10, Issue 8: 19058-19078. doi: 10.3934/math.2025852

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Research article

Fuzzy topology with $ \mathcal{I} $-convergence

1.
Laboratory of Mathematics and Applications to Engineering Sciences, École Normale Supérieure of Fez, Sidi Mohamed Ben Abdellah University, Fez, Morocco
2.
Laboratory of Sciences, Engineering and Management, Higher School of Technology of Fez, Sidi Mohamed Ben Abdellah University, Fez, Morocco
3.
Laboratory of Sustainable Agriculture Management, Higher School of Technology of Sidi bennour, Chouaib Doukkali University, El Jadida, Morocco
4.
Department of Mathematics, Indian Institute of Information Technology Sonepat, Haryana, India
5.
Department of Information Technology, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 26 June 2025 Revised: 05 August 2025 Accepted: 08 August 2025 Published: 21 August 2025
MSC : 03E72, 26E50, 54A40, 46S40, 34K36

In this paper, a new approach to specifying the closure of subsets in $ J^{E} $ is proposed, based on the concept of $ \mathcal{I} $-convergence of sequences. The notion of fuzzy $ \mathcal{I} $-closure is defined using this convergence, and from it, the fuzzy $ \mathcal{I} $-topology is introduced. Moreover, the concepts of continuity and $ \mathcal{I} $-continuity in fuzzy topological spaces are examined. A comparison between the proposed $ \mathcal{I} $-topology and the classical fuzzy topology is presented, highlighting certain fundamental properties and structural differences. This work contributes to the generalization of fuzzy topological structures by extending the role of sequence convergence through the use of $ \mathcal{I} $-convergence. Furthermore, a practical example illustrating temperature control in an industrial furnace is provided through the application of $ \mathcal{I} $-convergence of fuzzy sequences.
- fuzzy $ \mathcal{I} $-closure,
- fuzzy $ \mathcal{I} $-interior,
- fuzzy $ \mathcal{I} $-topology,
- fuzzy $ \mathcal{I} $-continuity,
- asymptotic density,
- $ t $-norm
Citation: Abdelhak Razouki, Omar El Ogri, Jaouad EL-Mekkaoui, Zamir Ahmad Ansari, Naglaa F. Soliman, Abeer D. Algarni. Fuzzy topology with $ \mathcal{I} $-convergence[J]. AIMS Mathematics, 2025, 10(8): 19058-19078. doi: 10.3934/math.2025852

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Abstract

In this paper, a new approach to specifying the closure of subsets in $ J^{E} $ is proposed, based on the concept of $ \mathcal{I} $-convergence of sequences. The notion of fuzzy $ \mathcal{I} $-closure is defined using this convergence, and from it, the fuzzy $ \mathcal{I} $-topology is introduced. Moreover, the concepts of continuity and $ \mathcal{I} $-continuity in fuzzy topological spaces are examined. A comparison between the proposed $ \mathcal{I} $-topology and the classical fuzzy topology is presented, highlighting certain fundamental properties and structural differences. This work contributes to the generalization of fuzzy topological structures by extending the role of sequence convergence through the use of $ \mathcal{I} $-convergence. Furthermore, a practical example illustrating temperature control in an industrial furnace is provided through the application of $ \mathcal{I} $-convergence of fuzzy sequences.

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