α-Topological Operators on Fuzzy Soft Sets and Applications to Hierarchical Clustering

Threshold-based [Formula: see text]-interior and [Formula: see text]-closure operators are introduced on fuzzy soft sets, acting directly on membership values via a pointwise threshold rule without presupposing any topological structure. These operators are proved to satisfy the standard axiomatic characterizations of interior and closure operators, and to jointly generate a coherent fuzzy soft topological space under the duality [Formula: see text], in which the complement of every open fuzzy soft set is closed. The practical utility of the framework is demonstrated in two settings. First, a medical diagnosis example shows that the intersection axiom of the induced topology directly identifies high-risk patients, the parameter [Formula: see text] serving as a clinician-controlled decision threshold and the boundary cases flagged automatically. Second, a novel sample-based hierarchical clustering method is proposed, in which a [Formula: see text] topological interaction matrix is constructed for each sample and each pair of parameters, encoding pairwise overlaps between the [Formula: see text]-interior, [Formula: see text]-closure, [Formula: see text]-boundary, and complement-boundary representations. Ward hierarchical clustering with Paik–Mondal similarity is applied to the resulting feature vectors and evaluated on four benchmark datasets from the UCI Machine Learning Repository, achieving clustering accuracies of [Formula: see text] (Iris), [Formula: see text] (Wine), [Formula: see text] (Breast Cancer Wisconsin), and [Formula: see text] (Glass Identification).