Complex-Valued Fuzzy Topological Spaces: Theory and Seasonal Pattern Prediction

A new approach based on the concept of fuzzy spaces (F-spaces) for the study of fuzzy topological spaces was developed by Dib in 1999. Extending that line of inquiry, a subset of the current authors previously investigated further topological notions, including boundary, exterior, isolated points and separation axioms, within Dib’s fuzzy-space framework (Jawarneh et al., 2021). In the present paper, we advance this program by introducing the concept of a complex fuzzy topological space (CFT-space), obtained by extending the codomain of the membership function of the fuzzy topological space from [0, 1] to the unit disk in the complex plane. This extension provides greater scope and flexibility for representing objects with simultaneous uncertainty and periodic semantics while preserving the full informational content. Basic operations on complex fuzzy spaces (CF-spaces), equality, difference, intersection and union are investigated within this new framework. Furthermore, we introduce the notion of a neighbourhood of a complex fuzzy point (CF-point), and we develop the concepts of the interior and closure of a complex fuzzy subspace (CF-subspace) together with its properties and associated results. To demonstrate the practical utility of our theoretical framework, we present an illustrative example for seasonal pattern prediction in environmental monitoring systems, which illustrate how the phase component naturally encodes temporal periodicity while the topological operators provide powerful tools for pattern analysis and prediction that are unavailable in classical fuzzy approaches.