Over the past sixty years, the study of metric properties lying between compactness and completeness has significantly enriched classical analysis. In an effort to extend these investigations beyond metric spaces, this treatise explores two such intermediary properties-namely, the Lebesgue property and weak G-completeness-in various ramifications of topological and metric spaces. Beginning with a historical backdrop, it examines these properties in fuzzy metric spaces and further broadens their scope to generalized topological settings, including generalized quasi-uniform spaces and g-quasi...