我们结合一个理想拓扑空间$(X, τ, \mathcal{I})$与一个范围函数$\mathfrak{a}: X → τ$，$x ∈ \mathfrak{a}(x)$，形成我们称之为理想-aura拓扑空间$(X, τ, \mathcal{I}, \mathfrak{a})$。核心的新对象是aura局部函数$A^{\mathfrak{a}}(\mathcal{I}) = \{x ∈ X : \mathfrak{a}(x) ∩ A ∉ \mathcal{I}\}$，它扩展了Jankovic-Hamlett局部函数：我们总是有$A^{*}(\mathcal{I}, τ) ⊆ A^{\mathfrak{a}}(\mathcal{I})$。闭包$\operatorname{cl}^{*}_{\mathfrak{a}}(A) = A ∪ A^{\mathfrak{a}}(\mathcal{I})$是一个加法性Cech闭包算子，一般来说，它不满足幂等性；我们证明幂等性等价于$\mathfrak{a}$的传递性。所得到的Cech拓扑$τ^{*}_{\mathfrak{a}}$处于链$τ_{\mathfrak{a}} ⊆ τ^{*}_{\mathfrak{a}} ⊆ τ^{*}$中，介于纯aura拓扑和经典理想拓扑之间。我们引入了一个$ψ_{\mathfrak{a}}$-算子，并用它来给出$τ^{*}_{\mathfrak{a}}$的另一种描述。定义了五类$\mathcal{I}\mathfrak{a}$-广义开集，并按层次排列，严格包含关系由反例分隔。证明了$\mathcal{I}\mathfrak{a}$-连续性的分解定理。特别研究了三种特殊情况：平凡理想恢复纯aura拓扑，不恰当理想给出离散拓扑，而有限集的理想表现出局部化现象。
英文摘要：
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.
School of Mathematics and Computational Science, Wuyi University（五邑大学数学与计算科学学院）；School of Mathematical Sciences, Guangxi Minzu University（广西民族大学数学科学学院）
