INTERVAL-VALUED OPEN SETS VIA MINIMAL AND MAXIMAL IN INTERVAL-VALUED TOPOLOGICAL SPACES

This study aims to develop and investigate several classes of interval-valued sets within the framework of Interval-Valued topology. Interval-Valued sets, originally proposed by Yao and later enriched through various generalizations, provide a more expressive structure for modelling uncertainty than classical or fuzzy sets. In this paper, we introduce and analyse different categories of Interval-Valued sets, particularly focusing on their weak and strong forms, and we explore how these forms influence topological behaviour. Special emphasis is placed on the study of minimal and maximal Interval-Valued open sets, pre-open sets, and semi-open sets, which serve as extremal elements in the lattice of Interval-Valued topologies. We also examine the interrelationships between various kinds of Interval-Valued closed sets and their generalized counterparts, thereby clarifying how such structures are embedded within the hierarchy of Interval-Valued topology. By establishing these connections, we provide a deeper understanding of the internal organisation of generalized closed sets in this setting. To support the theoretical results, illustrative and carefully constructed examples are presented, which highlight both the necessity and the distinctions among the introduced concepts. Overall, this work contributes to the enrichment of Interval-Valued topological theory and lays the foundation for further applications in mathematical modelling and decision-making under uncertainty.