A holistic perspective on soft separation axioms: Addressing open problems, rectifications, and introducing novel classifications

Soft topology establishes its importance as a frame of reference through numerous formulas derived for each classical topological concept, implying that classical topology is a special case obtained when the set of parameters is a singleton. In this work, we successfully solve two open problems concerning the relationships between two types of soft separation axioms in two categories. Then, we amend existing example showing that soft TB 0 ? soft TS 0. In this context, we clarify that soft TB0? soft TS 0 if the set of parameters is finite. In contrast, we construct a soft topological structure with infinite set of parameters to illustrate that soft TB0 ? soft TS0. Finally, we define a new form of soft points inspired by fuzzy points. Surprisingly, the new definition results in a spectrum of soft points that starts at ?x and ends at (x,P) for every x ? U, where P is the set of parameters andU is the universe. We make use of this sort of soft points to create two classes of separation axioms via soft topologies: {soft T0, soft T1, soft T2, soft T3, soft T4} and {soft T00, soft T01, soft T02, soft T03, soft T04}. The master features of these axioms are scrutinized and the relationships between them as well as their relationships with the foregoing ones are revealed with the help of interesting counterexamples. Especially, we clarify that the axioms of soft TS and soft TE structures are special case of the current classes. Among the interesting results that we obtain are the identity between soft T3 and classical T3 structures and the equivalence between soft T1 and soft TB1 structures.