Soft aura topological spaces and rough approximation operators

Abstract: We introduce the concept of a soft aura topological space $(X, \tildeτ, \mathfrak{a}_E)$, obtained by equipping a soft topological space $(X, \tildeτ, E)$ with a soft scope function $\mathfrak{a}_E : X \to \tildeτ$ satisfying $x \in \mathfrak{a}_E(x)(e)$ for every $x \in X$ and every parameter $e \in E$. This framework generalizes the recently introduced aura topological spaces to the soft setting. We define the soft aura-closure operator and the soft aura-interior operator, and prove that the closure is a soft additive Čech closure operator whose transfinite iteration yields a soft Kuratowski closure. Five classes of generalized soft open sets -- soft $\mathfrak{a}$-semi-open, soft $\mathfrak{a}$-pre-open, soft $\mathfrak{a}$-$α$-open, soft $\mathfrak{a}$-$β$-open, and soft $\mathfrak{a}$-$b$-open sets -- are introduced, and a complete hierarchy among them is established. Soft $\mathfrak{a}$-continuity and its decompositions are studied. Separation axioms soft $\mathfrak{a}$-$T_i$ ($i = 0, 1, 2, 3$) are introduced; it is shown that soft $\mathfrak{a}$-$T_1$ and soft $\mathfrak{a}$-$T_2$ coincide due to the scope-based formulation. Soft aura-based lower and upper rough approximation operators are defined, generalizing both the crisp aura rough set model and the classical Pawlak model. An illustrative application to environmental risk assessment demonstrates the practical utility of the proposed framework.

• The paper defines and explores soft aura topological spaces.
• The paper outlines five classes of generalized soft open sets and defines soft aura-closure with the interior operator that is additive with respect to unions
• The paper describes a use case example for environmental monitoring

Overview

This paper extends the author's recently introduced framework of aura topological spaces to the soft set setting of Molodtsov. A soft aura topological space is a quadruple $(X, \tau, a_E, E)$, where $(X, \tau, E)$ is a soft topological space in the sense of Shabir and Naz and $a_E : X \to \tau$ is a soft scope function assigning to each point $x$ a soft open neighborhood $a_E(x)$ satisfying the membership axiom $x \in a_E(x)(e)$ for every parameter $e \in E$. The construction generalizes the crisp aura space $(X, \tau, a)$, which is recovered when $E$ is a singleton. The paper develops four interlocking bodies of results: operator theory (closure/interior), generalized open sets and continuity, separation axioms, and rough approximation with an applied case study.

The central structural novelty is parameter-dependence: because each scope value $a_E(x)$ is an entire family $(X, \tau, E)$0 of subsets, the "scope" of a point can change shape from one parameter to another. This permeates every notion in the paper—operators act parameterwise, openness conditions may hold at some parameters but fail at others, and separation axioms acquire an existential-versus-universal quantification over $(X, \tau, E)$1 that has no crisp analogue.

Closure and interior operators

For $(X, \tau, E)$2, the soft aura-closure and interior are defined parameterwise:

$(X, \tau, E)$3

The paper proves that $(X, \tau, E)$4 satisfies grounding, enlargement, monotonicity, and full soft additivity, hence is a soft additive Čech closure operator. Crucially, $(X, \tau, E)$5 is not idempotent in general ($(X, \tau, E)$6 may differ from $(X, \tau, E)$7), mirroring the crisp aura phenomenon. The operators are dual via complementation: $(X, \tau, E)$8 and conversely.

Two derived structures follow. First, the collection $(X, \tau, E)$9 of fixed points of $a_E : X \to \tau$0 (soft $a_E : X \to \tau$1-open sets) forms a genuine soft topology; however, unlike the crisp case where $a_E : X \to \tau$2, for $a_E : X \to \tau$3 the relationship between $a_E : X \to \tau$4 and $a_E : X \to \tau$5 depends on the interplay between the scope function and the parameterization—an assumption-dependent point the paper states explicitly. Second, transfinite iteration of $a_E : X \to \tau$6 stabilizes at some ordinal $a_E : X \to \tau$7 (by cardinality of the increasing chains in each fiber), and the limit operator $a_E : X \to \tau$8 is a soft Kuratowski closure, generating a topology $a_E : X \to \tau$9. This repair mechanism—replacing the non-idempotent Čech closure by its transfinite stabilization—is used repeatedly later, notably in the continuity decomposition theorem.

Generalized open sets and continuity

Five classes of generalized soft open sets are introduced: soft $x$0-semi-open ($x$1), soft $x$2-pre-open ($x$3), soft $x$4-$x$5-open, soft $x$6-$x$7-open, and soft $x$8-$x$9-open. The paper establishes the complete hierarchy

$a_E(x)$0

with all implications strict. Arbitrary unions of semi-, pre-, and $a_E(x)$1-open sets remain in their respective classes. A limitation inherited from Čech non-idempotency is noted: finite intersections of soft $a_E(x)$2-$a_E(x)$3-open sets need not be $a_E(x)$4-$a_E(x)$5-open; this property is restored when working with $a_E(x)$6 instead of $a_E(x)$7.

On the mapping side, soft $a_E(x)$8-continuity and its semi-, pre-, $a_E(x)$9-, and $x \in a_E(x)(e)$0-variants inherit the same hierarchy. The main result here is a decomposition theorem: with respect to the Kuratowski closure $x \in a_E(x)(e)$1, soft $x \in a_E(x)(e)$2-$x \in a_E(x)(e)$3-continuity holds if and only if the map is both soft $x \in a_E(x)(e)$4-semi-continuous and soft $x \in a_E(x)(e)$5-pre-continuous. The proof exploits idempotency of $x \in a_E(x)(e)$6; the paper concedes that for the raw Čech operator $x \in a_E(x)(e)$7 the decomposition may fail. Additional characterizations include closure under composition and the standard closure-inequality characterization of continuity, $x \in a_E(x)(e)$8.

Separation axioms

The axioms soft $x \in a_E(x)(e)$9-$e \in E$0, $e \in E$1, $e \in E$2, regularity, and $e \in E$3 are formulated through the scope function itself rather than through arbitrary neighborhoods. Two findings stand out:

• Collapse of $e \in E$4 into $e \in E$5: soft $e \in E$6-$e \in E$7 holds iff $e \in E$8 for all $e \in E$9 and $(X, \tau, a)$0, which immediately forces disjoint scopes, so soft $(X, \tau, a)$1-$(X, \tau, a)$2 soft $(X, \tau, a)$3-$(X, \tau, a)$4. The paper attributes this collapse to the scope-based formulation per se—it operates identically in the crisp setting—and does not claim it as a distinctively soft effect.
• Widened $(X, \tau, a)$5–$(X, \tau, a)$6 gap: soft $(X, \tau, a)$7-$(X, \tau, a)$8 requires only that some parameter separates a given pair, whereas $(X, \tau, a)$9 requires separation at every parameter simultaneously. An explicit two-point example over $E$0 with the discrete soft topology realizes $E$1 but not $E$2, illustrating that scopes may carry different geometric information at different parameters.

Separation properties also depend strongly on the choice of scope function on a fixed underlying soft topological space: the trivial scope $E$3 yields a space failing even $E$4, while sufficiently rich $E$5 permits singleton scopes yielding $E$6. Under $E$7, singletons are closed in the aura sense, $E$8.

Rough approximation and application

Identifying lower approximation with $E$9 and upper approximation with $a_E(x)$0, the paper defines soft aura rough approximations, boundary, and an accuracy measure

$a_E(x)$1

and verifies the standard package: sandwiching, monotonicity, additivity of upper/lower over unions/intersections, duality, and $a_E(x)$2 with equality iff the boundary is null. When scopes arise from an equivalence relation (constant across parameters), the model reduces exactly to Pawlak's; thus the framework dispenses with the equivalence-relation requirement while adding multi-criteria resolution.

The application classifies five environmental monitoring stations against four indicators (PM$a_E(x)$3, SO$a_E(x)$4, pH, dissolved oxygen). The target "at-risk" soft set achieves $a_E(x)$5, indicating substantial boundary uncertainty. Stations $a_E(x)$6 (PM$a_E(x)$7) and $a_E(x)$8 (DO) fall in the lower approximation and are definitively at risk; pH and SO$a_E(x)$9 have empty lower approximations, identifying those indicators as the dominant sources of classification uncertainty—a directly actionable diagnostic for resource allocation. The numerical result is modest in scale (an illustrative example), and its practical utility rests on the interpretability of parameter-wise boundaries rather than on comparative benchmarking against other soft rough set models, which the paper does not undertake.

Limitations and open questions

Several caveats are stated within the paper itself. The Čech nature of $(X, \tau, E)$00 breaks idempotency, invalidating finite-intersection stability for $(X, \tau, E)$01-open sets and the continuity decomposition unless one passes to $(X, \tau, E)$02; whether the stabilized topology $(X, \tau, E)$03 retains enough of the original structure for applications is not examined. The containment relation between $(X, \tau, E)$04 and $(X, \tau, E)$05 for $(X, \tau, E)$06 is left dependent on the specific scope function. The equivalence $(X, \tau, E)$07 means the finer gradations of Hausdorff-type separation familiar from classical topology are unavailable in this framework. The application is a small worked example without empirical validation or comparison to existing soft rough decision models. The conclusion lists open problems: soft aura-compactness and connectedness, enrichment by a soft ideal, extensions to intuitionistic fuzzy soft and neutrosophic soft settings, and systematic decision-making algorithms built on the approximation operators.

Conclusion

The paper delivers a coherent soft-set lift of the aura framework: a parameterwise additive Čech closure with a Kuratowski stabilization, a strict five-class hierarchy of generalized open sets with matching continuity notions and a decomposition theorem, scope-based separation axioms exhibiting a collapsed $(X, \tau, E)$08 level and a widened $(X, \tau, E)$09 gap driven by parameter quantification, and Pawlak-generalizing rough approximations illustrated on an environmental risk problem. The theory's distinctive feature—the existential-versus-universal behavior of properties across the parameter set—is identified as the source of genuinely new phenomena relative to the crisp model, while the reliance on transfinite stabilization to recover classical properties remains the framework's principal technical dependency.