Soft Bitopological Spaces via Soft Elements

We introduce soft bitopological spaces from the standpoint of soft elements. A soft bitopological space is a soft set equipped with two soft topologies. Following the classical construction of Goldar–Ray, each soft topology on $F$ induces an ordinary topology on the set $\SE(F)$ of soft elements; hence every soft bitopological space canonically determines a genuine bitopological space on $\SE(F)$. Within this setting we define pairwise soft separation axioms ($T_0$, $T_1$, $T_2$) and a notion of pairwise soft compactness, and we compare them with their parameterwise counterparts. For canonical (sectionwise generated) soft bitopologies, we show that the pairwise soft $T_i$ axioms are equivalent to the corresponding pairwise $T_i$ axioms on each parameter space. Compactness exhibits a finiteness phenomenon: when the parameter set is finite, componentwise pairwise compactness forces pairwise soft compactness, while an infinite-parameter example shows that the finiteness assumption is essential. Examples are included to clarify how the induced bitopology on $\SE(F)$ may behave differently from the original soft bitopology.

💡 Research Summary

The paper introduces a novel framework called a “soft bitopological space,” which consists of a soft set F together with two soft topologies τ₁ and τ₂ defined on it. The authors adopt the soft‑element viewpoint pioneered by Goldar and Ray: a soft element is a selection a : A → X such that a(t) ∈ F(t) for every parameter t in the fixed parameter set A. For any soft topology τ on F, the collection of soft elements whose sections belong to the corresponding component topologies yields a classical topology τ* on the set SE(F) of all soft elements. Consequently, a soft bitopological space (F, τ₁, τ₂) canonically determines a genuine bitopological space (SE(F), τ₁*, τ₂*).

The paper first revisits basic notions of soft sets, soft elements, and the section operator that maps a family of soft elements to a soft set. It then defines “canonical” soft topologies: given a family of ordinary topologies {σₜ}ₜ∈A on the parameterwise sections F(t), the set Top({σₜ}) consists of all soft subsets H of F such that H(t) ∈ σₜ for every t. This construction yields a soft topology that is the smallest enlargement of any given soft topology sharing the same component topologies. The authors show that the induced classical topologies τ* depend only on the component topologies, so the canonical enlargement does not alter the induced bitopology.

The central contributions are the definitions of pairwise separation axioms and pairwise compactness for soft bitopological spaces, expressed in terms of soft elements:

• Pairwise soft T₀: For any distinct soft elements a, b there exists a soft open set belonging to either τ₁ or τ₂ that contains one of them but not the other.
• Pairwise soft T₁: For any distinct a, b there are H ∈ τ₁ and K ∈ τ₂ such that a ∈ H, b ∉ H and b ∈ K, a ∉ K.
• Pairwise soft T₂ (Hausdorff): For any distinct a, b there exist H ∈ τ₁ and K ∈ τ₂ with a ∈ H, b ∈ K and H ∩ K = ∅ (in the soft‑set sense).

Proposition 4.4 establishes the usual implication chain T₂ ⇒ T₁ ⇒ T₀. Theorem 4.5 provides a componentwise characterization for canonical soft bitopologies: a soft bitopological space satisfies a pairwise Tⱼ (j = 0, 1, 2) if and only if each parameter space (F(t), (τ₁)ₜ, (τ₂)ₜ) satisfies the corresponding classical pairwise Tⱼ. The proof constructs soft elements that agree on all parameters except the distinguished one, forcing the separating open set to act at that parameter.

Compactness is treated analogously. A soft bitopological space is pairwise soft compact if every cover by soft open sets from τ₁ ∪ τ₂ admits a finite subcover. Theorem 5.2 shows that when the parameter set A is finite, componentwise pairwise compactness (i.e., each (F(t), (τ₁)ₜ, (τ₂)ₜ) is pairwise compact) implies pairwise soft compactness of the whole space. An explicit infinite‑parameter counterexample demonstrates that finiteness is essential: with infinitely many parameters, componentwise compactness does not guarantee soft compactness.

The paper includes several illustrative examples. One shows that a soft open set H may induce a classical open set H* that fails to separate the corresponding soft elements, highlighting that the induced bitopology on SE(F) can behave differently from the original soft bitopology. Another example constructs a canonical soft bitopology whose induced topologies coincide with the component topologies, confirming the theoretical results. The final examples emphasize the finiteness phenomenon for compactness and the non‑equivalence of separation properties between the soft and induced classical settings.

In the concluding section, the authors suggest further research directions, such as extending soft bitopological concepts to quasi‑metric spaces, ordered topologies, and soft measure theory, where pairwise notions naturally arise. By integrating soft set theory with classical bitopology via the soft‑element perspective, the paper provides a robust platform for transferring and generalizing many topological concepts to the soft context, opening avenues for applications in areas where information is parameter‑dependent and possibly incomplete.