Affiliations: Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt
Correspondence:
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Corresponding author. Ahmed Mostafa Khalil, Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt. Tel.: +2010280282950. E-mail: [email protected].
Abstract: In this article, we will define the new notions (e.g., semi-θ-neighbor-hood system of point, semi-θ-closure (interior) of a set and semi-θ-closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of above five notions in detail. Several basic results (for instance, Definition 2.3, Theorem 2.5 (iii), (v) and (vi), Theorem 2.10, Theorem 2.14 and Theorem 4.6) in classical topology are generalized to the fuzzy case based on Łukasiewicz logic. In addition to, we will show that every fuzzifying semi-θ-closed set is fuzzifying semi-closed set (by Theorem 2.5 (vi)). Further, we will study the notion of fuzzifying semi-θ-derived set and fuzzifying semi-θ-boundary set, and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly semi-θ-continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly semi-θ-continuous mapping, fuzzifying strongly semi-θ-irresolute mapping, and fuzzifying weakly semi-θ-irresolute mapping along with different conditions for their existence are obtained.
Keywords: Fuzzy logic, fuzzifying topology, fuzzifying semi-θ-closure of a set, fuzzifying semi-θ-closed sets, fuzzifying strongly semi-θ-continuous mapping
