发表机构
The University of Burdwan（布尔达万大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出统一的 $Q\text{--}P$ 空间框架，涵盖广义拓扑结构，建立保持性结果，并研究中间 Hausdorff 类，揭示从 $T_2$ 到 $T_1$ 的严格层次关系及与多种拓扑性质的关联。
AI 中文摘要
我们引入了一个统一的 $Q\text{--}P$ 空间框架，旨在涵盖由两个拓扑性质相互作用而产生的一类广泛的广义拓扑结构。我们建立了普遍的保持性和传递性结果，包括遗传性、可积性、可加性和分离性质。专门针对 Hausdorff 情形，我们研究了中间类 $C$-Hausdorff、$PS$-Hausdorff 和 $CC$-Hausdorff，并得到了层次关系 \\[ T_2 \Longrightarrow Ps\text{-}Hausdorff \Longrightarrow CC\text{-}Hausdorff \Longrightarrow C\text{-}Hausdorff \Longrightarrow T_1. \\] 对于 $C$-Hausdorff 空间，我们推导了内在刻画，并建立了其与 $k$-Hausdorff 性和 $KC$ 性同时成立的等价性。我们进一步考察了它们与 $k$-空间、Fréchet 空间、局部紧性、实紧性以及 $\aleph_0$-有界性的联系。最后，精心构造的例子和反例证明了层次关系的严格性以及假设的必要性。因此，$Q\\!-\\!P$ 框架为广义分离公理提供了一个统一的视角，并揭示了经典 Hausdorff 性与 $T_1$-分离之间的丰富结构。
英文摘要
We introduce a unified framework of $Q\text{--}P$ spaces, designed to encompass a broad class of generalized topological structures arising from the interaction of two topological properties. We establish general permanence and transfer results, including heredity, productivity, additivity, and separation properties. Specializing to the Hausdorff setting, we investigate the intermediate classes \(C\)-Hausdorff, \(PS\)-Hausdorff, and \(CC\)-Hausdorff, and obtain the hierarchy \[ T_2 \Longrightarrow Ps\text{-}Hausdorff \Longrightarrow CC\text{-}Hausdorff \Longrightarrow C\text{-}Hausdorff \Longrightarrow T_1. \] For \(C\)-Hausdorff spaces, we derive intrinsic characterizations and establish the equivalence with the simultaneous properties of \(k\)-Hausdorffness and \(KC\). We further examine their connections with \(k\)-spaces, Fréchet spaces, local compactness, realcompactness, and \(\aleph_0\)-boundedness. Finally, carefully constructed examples and counterexamples demonstrate the strictness of the hierarchy and the necessity of the hypothesis. Thus, the \(Q\!-\!P\) framework provides a unified perspective on generalized separation axioms and reveals a rich structure between classical Hausdorffness and \(T_1\)-separation.
