Malaya Journal of Matematik, Vol. 8, No. 3, 1114-1118, 2020 https://doi.org/10.26637/MJM0803/0065 On fuzzy inverse systems of fuzzy topological spaces B. M. Uzzal Afsan * Abstract In this paper, we have initiated and studied the concepts of fuzzy inverse systems of fuzzy topological spaces and fuzzy continuous functions along with their fuzzy inverse limits. We have shown that every fuzzy inverse system possesses fuzzy inverse limit and fuzzy inverse limit is unique in some sense. The fuzzy inverse limit (X , ψ i ) I of a fuzzy inverse system (X i , ψ ij ) I enjoys many satisfactory desired properties. Further, we have introduced a covariant functor F lim ← from the category FIS(I ) of fuzzy inverse systems of fuzzy topological spaces and fuzzy continuous functions to the category FTS of fuzzy topological spaces and fuzzy continuous functions. Keywords Fuzzy inverse system, fuzzy inverse limit, category, covariant functor. AMS Subject Classification 26A33, 30E25, 34A12, 34A34, 34A37, 37C25, 45J05. Department of Mathematics, Sripat Singh College, Jiaganj-742123, Murshidabad, West Bengal, India. *Corresponding author: uzlafsan@gmail.com Article History: Received 20 April 2020; Accepted 16 July 2020 ©2020 MJM. Contents 1 Introduction ...................................... 1114 2 Preliminaries ..................................... 1114 3 Fuzzy inverse systems of fuzzy topological spaces 1115 4 Category of fuzzy inverse systems of fuzzy topologi- cal spaces and fuzzy continuous functions ..... 1117 5 Conclusion ....................................... 1118 References ....................................... 1118 1. Introduction In 1965, Zadeh [10] generalized the notion of “sets” to “fuzzy sets”, which was a great achievement not only in pure mathematics, but also in the whole world of mathematical sciences because it has several direct applications in different branches of science. “Fuzzy topology”, initiated by Chang [3], becomes a mature field in fuzzy mathematics. To build the foundation of fuzzy topology, the works of Lowen [6, 7], Pao-Ming and Ying-Ming [8], Guojun [5] are worths to be mentioned. The notions of inverse systems and their inverse limits has several application in different branches of mathematics, specially in category theory [4] and finite group theory [9]. The main purpose of this article is to introduce the concepts of fuzzy inverse systems of fuzzy topological spaces and fuzzy continuous functions and their fuzzy inverse limits. In section 3, We have achieved a bridge result between the fuzzy inverse system of the topological spaces and the inverse system of the fuzzy topological spaces. We have also established that every fuzzy inverse system of fuzzy topological spaces and fuzzy continuous functions has fuzzy inverse limit. Further we have shown that every fuzzy inverse system (X i , ψ ij ) I possesses the unique fuzzy inverse limit in the sense that if (X , ψ i ) I and (X ′ , ψ ′ i ) I are two fuzzy inverse limits of (X i , ψ ij ) I , then there exists a fuzzy homeomorphism F : X → X ′ such that ψ ′ i F = ψ i for each i ∈ I . We have studied several other basic properties of the notion of fuzzy inverse limit (X , ψ i ) I of a fuzzy inverse system (X i , ψ ij ) I . Besides these, in section 4, we have initiated a new covariant functor F lim ← from the category FIS(I ) to the category FTS of fuzzy topological spaces and fuzzy continuous functions. 2. Preliminaries Zadeh [10] initiated the concept of “fuzzy set”. Definition 2.1. Suppose X be a non-empty set. Then any function with domain X and codomain J =[0, 1] is said to be a fuzzy subset of the set X and the collection of all fuzzy subsets of the X is denoted by J X .