arXiv subjects

Ahu Acikgoz

Publications and source records attributed to Ahu Acikgoz.

4 recordsLinked to original sources

arXiv2026

Ideal-Aura Topological Spaces, New Local Functions, and Generalized Open Sets

We combine an ideal topological space $(X, τ, \mathcal{I})$ with a scope function $\mathfrak{a}: X \to τ$, $x \in \mathfrak{a}(x)$, to form what we call an ideal-aura topological space $(X, τ, \mathcal{I}, \mathfrak{a})$. The central new object is the aura-local function $A^{\mathfrak{a}}(\mathcal{I}) = \{x \in X : \mathfrak{a}(x) \cap A \notin \mathcal{I}\}$, which extends the Jankovic-Hamlett local function: we always have $A^{*}(\mathcal{I}, τ) \subseteq A^{\mathfrak{a}}(\mathcal{I})$. The closure $\operatorname{cl}^{*}_{\mathfrak{a}}(A) = A \cup A^{\mathfrak{a}}(\mathcal{I})$ is an additive Cech closure operator that, in general, fails to be idempotent; we prove that idempotency is equivalent to transitivity of $\mathfrak{a}$. The resulting Cech topology $τ^{*}_{\mathfrak{a}}$ sits in the chain $τ_{\mathfrak{a}} \subseteq τ^{*}_{\mathfrak{a}} \subseteq τ^{*}$, interpolating between the pure aura topology and the classical ideal topology. We introduce a $ψ_{\mathfrak{a}}$-operator and use it to give an alternative description of $τ^{*}_{\mathfrak{a}}$. Five classes of $\mathcal{I}\mathfrak{a}$-generalized open sets are defined and arranged in a hierarchy, with strict inclusions separated by counterexamples. Decomposition theorems for $\mathcal{I}\mathfrak{a}$-continuity are proved. Three special cases are examined: the trivial ideal recovers the pure aura topology, the improper ideal gives the discrete topology, and the ideal of finite sets exhibits a localization phenomenon.

math.GN↗

arXiv2026

Soft aura topological spaces and rough approximation operators

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.

math.GN↗

arXiv2026

Aura Topological Spaces and Generalized Open Sets with Applications to Rough Sets, Sensor Networks, and Epidemic Modelling

We equip a topological space $(X,τ)$ with a function $\mathfrak{a}: X \to τ$ satisfying the single axiom $x \in \mathfrak{a}(x)$. The resulting triple $(X, τ, \mathfrak{a})$, which we call an aura topological space, provides a point-to-open-set assignment that differs from all existing auxiliary structures in topology. The aura-closure operator $\text{cl}_{\mathfrak{a}}(A) = \{x \in X : \mathfrak{a}(x) \cap A \neq \emptyset\}$ turns out to be an additive Cech closure operator; it satisfies extensivity, monotonicity, and finite additivity, but idempotency fails in general. Iterating $\text{cl}_{\mathfrak{a}}$ transfinitely yields a Kuratowski closure whose topology $τ_{\mathfrak{a}}^{\infty}$ satisfies $τ_{\mathfrak{a}}^{\infty} \subseteq τ_{\mathfrak{a}} \subseteq τ$. We introduce five classes of generalized open sets, determine their complete hierarchy, and separate all non-coinciding classes by counterexamples. Continuity notions, decomposition theorems, and separation axioms are studied. Three applications are developed: rough set approximations generalizing Pawlak's model, wireless sensor network coverage analysis, and epidemic spread modelling.

math.GN↗

arXiv2026

Compactness and Connectedness in Aura Topological Spaces

This is the second paper in a series on aura topological spaces $(X, τ, \mathfrak{a})$, where $\mathfrak{a}: X \to τ$ is a scope function with $x \in \mathfrak{a}(x)$. We study covering and connectivity properties in this setting. Five compactness-type notions are defined ($\mathfrak{a}$-compact, $\mathfrak{a}$-Lindelof, countably $\mathfrak{a}$-compact, $\mathfrak{a}$-sequentially compact, $\mathfrak{a}$-limit point compact) and their mutual relationships are determined. For transitive aura functions we obtain a concrete convergence criterion: $(x_n)$ converges to $x$ in $τ_{\mathfrak{a}}$ if and only if $x_n \in \mathfrak{a}(x)$ eventually. We show that $\mathfrak{a}$-compact subsets of $\mathfrak{a}$-$T_2$ spaces are $\mathfrak{a}$-closed and that $\mathfrak{a}$-compactness is preserved under $\mathfrak{a}$-continuous surjections. On the connectivity side, $\mathfrak{a}$-connected, $\mathfrak{a}$-path connected, and $\mathfrak{a}$-locally connected spaces are introduced; $\mathfrak{a}$-components are $\mathfrak{a}$-closed, and they are $\mathfrak{a}$-open when the space is $\mathfrak{a}$-locally connected. We construct subspace and product aura topologies. For products the inclusion chain $(τ_{\mathfrak{a}}) \times (τ_{\mathfrak{b}}) \subseteq τ_{\mathfrak{a} \times \mathfrak{b}} \subseteq τ_X \times τ_Y$ is established, with equality on the left when both scope functions are transitive. A Tychonoff-type theorem for transitive aura spaces is proved. All implications are shown to be strict by counterexamples.

math.GN↗