On fuzzy compact and fuzzy connected spaces defined on a fuzzy set

This research investigates the structure and relationships among different types of fuzzy compact and fuzzy connected spaces defined over fuzzy topological spaces. It introduces and analyzes three primary types of fuzzy compactness—fuzzy regular g-compact (f.RG.co), fuzzy g-compact (f.G.co), and fuzzy g*-compact (f.G*.co)—each based on different generalized open sets. It also explores four types of fuzzy connectedness—fuzzy rg-connected, g-connected, g*-connected, and fuzzy connected—defined through various fuzzy separation concepts. The study uses a deductive theoretical approach to formally define each space type, prove implication relationships among them, and construct logical hierarchies. Results show that every f.RG.co space implies f.G.co, which in turn implies f.G*.co, all of which imply fuzzy regular compactness. Similarly, fuzzy rg-connectedness implies g-connectedness, which implies g*-connectedness, which implies classical fuzzy connectedness. The paper includes two diagrams to visualize the inclusion structure of compact and connected space types and provides two summary tables that compare their properties and logical implications.