A Holistic Perspective on Soft Separation Axioms: Addressing Open Problems, Rectifications, and Introducing Novel Classifications

Abstract

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 $\text{soft\;}T_0^B\not\rightarrow \text{soft\;}T_0^S$. In this context, we clarify that $\text{soft\;}T_0^B\rightarrow \text{soft\;} T_0^S$ if the set of parameters is finite. In contrast, we construct a soft topological structure with infinite set of parameters to illustrate that $\text{soft\;}T_0^B\not\rightarrow \text{soft\;}T_0^S$. 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 $\varepsilon_x$ and ends at $(x, \mathcal{P})$ for every $x \in \mathcal{U}$, where $\mathcal{P}$ is the set of parameters and $\mathcal{U}$ is the universe. We make use of this sort of soft points to create two classes of separation axioms via soft topologies: $\{\text{soft\;}T^{0},\text{soft\;}T^{1},\text{soft\;}T^{2},
\text{soft\;}T^{3},\text{soft\;}T^{4}\}$ and $\{\text{soft\;}T^{00},\text{soft\;}T^{01},\text{soft\;}T^{02},
\text{soft\;}T^{03},\text{soft\;}T^{04}\}$. 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 $T^S$ and soft $T^E$ structures are special case of the current classes. Among the interesting results that we obtain are the identity between soft $T^3$ and classical $T_3$ structures and the equivalence between soft $T^{1}$ and soft $T^{B}_{1}$ structures.