Aura Topological Spaces and Generalized Open Sets with Applications to Rough Sets, Sensor Networks, and Epidemic Modelling
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| Formato: | Preprint |
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2026
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| _version_ | 1866915784296497152 |
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| author | Acikgoz, Ahu |
| author_facet | Acikgoz, Ahu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_07678 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Aura Topological Spaces and Generalized Open Sets with Applications to Rough Sets, Sensor Networks, and Epidemic Modelling Acikgoz, Ahu General Topology 54A05, 54A10, 54C08, 54D10, 03E72, 94C15, 92D30 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. |
| title | Aura Topological Spaces and Generalized Open Sets with Applications to Rough Sets, Sensor Networks, and Epidemic Modelling |
| topic | General Topology 54A05, 54A10, 54C08, 54D10, 03E72, 94C15, 92D30 |
| url | https://arxiv.org/abs/2602.07678 |