aura拓扑空间中的Levine等价：可达性、Alexandrov结构与商框架
Levine Equivalence in Aura Topological Spaces: Reachability, Alexandrov Structure, and Quotient Frames
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中文总结 AI 辅助
本文研究aura拓扑空间中的Levine等价，证明其为Alexandrov拓扑，给出aura-Levine包的显式表示，刻画等价条件与商结构，对有限空间给出等价判定步骤。
中文摘要 AI 辅助
本文研究 aura拓扑空间中的Levine等价。证明 aura拓扑由范围函数诱导的可达性预序确定，因此是Alexandrov拓扑；给出aura-Levine包的显式表示$K_{\mathfrak{a}}(A)=\uparrow_{\mathfrak{a}}A=S_{\mathfrak{a}}^{\infty}(A)$，故两个子集aura-Levine等价当且仅当它们具有相同的最终前向传播。还描述等价类的商结构，通过可达性预序刻画分离性质，分类诱导相同Levine等价的范围函数，研究其在aura连续映射下的保真性；对有限aura空间，给出利用范围关系的自反传递闭包判定aura-Levine等价的显式步骤。
英文摘要
In this paper, we study Levine equivalence in aura topological spaces. We show that the aura topology is determined by the reachability preorder induced by the scope function and is therefore an Alexandrov topology. We prove that the aura-Levine hull has the explicit representation $K_{\mathfrak a}(A)=\uparrow_{\mathfrak a}A=S_{\mathfrak a}^{\infty}(A)$, and hence two subsets are aura-Levine equivalent if and only if they have the same eventual forward spread. We also describe the quotient structure of the equivalence classes, characterize separation properties through the reachability preorder, classify scope functions that induce the same Levine equivalence, and examine preservation under aura-continuous mappings. For finite aura spaces, we provide an explicit procedure for deciding aura-Levine equivalence using the reflexive-transitive closure of the scope relation.
