A Note on Hesitant Fuzzy Approximation Spaces and Hesitant Fuzzy Topological Spaces

A hesitant fuzzy element is defined as a discrete array characterized by complexity and diversity. Consequently, certain inadequate propositions have been identified in the existing literature on hesitant fuzzy sets. The development of a mathematical knowledge system relies on well-established theoretical foundations; even minor deficiencies in these foundations can undermine the integrity of subsequent knowledge frameworks. Topology and approximation space theory serve as essential pillars for a broad range of theoretical constructs. To rectify the flawed propositions in hesitant fuzzy approximation spaces and hesitant fuzzy topological spaces, this study proposes new intersection and union operators for hesitant fuzzy sets and refines the fundamental propositions of hesitant fuzzy sets under these operators. By presenting the inadequate propositions from the literature on these two mathematical structures and replacing the original intersection and union operations with the newly defined ones, all previously identified inadequacies can be rigorously addressed. Consequently, the theories derived from these propositions are also rigorously valid.