Combination of Fuzzy Vector Spaces and Fuzzy Topology: Foundations and Applications
Freddy Tsatsa Bakweno *
Sciences, Department of Mathematics and Computer Science, Higher Pedagogical Institute of Popokabaka, ISPP, DRC.
Huyghens N'sungu Kuleya
Exact Sciences, Department of Mathematics and Physics, Higher Pedagogical Institute of Kasongo, Lunda, ISPK, DRC.
Zéphirin N'teba Makala
Department of Mathematics and Computer Science, Faculty of Science and Technology, National Pedagogical University, Kinshasa, DRC.
Gérard Tawaba Musian Ta-yen
Department of Mathematics and Computer Science, Faculty of Science and Technology, National Pedagogical University, Kinshasa, DRC.
Rostin Mabela Makengo
Department of Mathematics and Computer Science, Faculty of Science and Technology, University of Kinshasa, Kinshasa, DRC.
*Author to whom correspondence should be addressed.
Abstract
This article proposes a unified mathematical framework combining the theory of fuzzy vector spaces and that of fuzzy topologies to model complex systems characterized by uncertainty and structural imprecision. We formally introduce the notion of a fuzzy topological vector space and study its fundamental algebraic and topological properties, including the continuity of vector operations, fuzzy separation axioms, and fuzzy compactness and connectivity. A constructive approach based on α-cuts is developed to establish a rigorous link between fuzzy structures and classical topological vector spaces. Furthermore, we extend this framework to fuzzy functional analysis, fuzzy Sobolev spaces, fuzzy partial differential equations, and fuzzy dynamical systems. Potential applications in optimal control, image processing, artificial intelligence, and mathematical physics are also discussed.
This work constitutes a theoretical contribution towards the coherent integration of algebra, topology and uncertainty, paving the way for the development of new mathematical tools for the analysis of complex systems. Future work could focus on integrating fuzzy topological vector spaces with fuzzy differential geometry, fuzzy manifolds, fuzzy neural networks, and advanced numerical methods, in order to develop a comprehensive analytical and computational theory.
Keywords: Fuzzy vector spaces, fuzzy topologies, fuzzy topological vector spaces, fuzzy functional analysis, mathematical uncertainty