Pervin 空间的无点理论
A pointfree theory of Pervin spaces
AI总结：
本文建立 Pervin 空间的无点理论，提出 Frith frame 范畴，证明其与 Pervin 空间的对偶伴随及在拟一致 frame 中的满全反射地位，并刻画其完备性与完备化。
摘要：
我们奠定了 Pervin 空间的无点理论基础。Pervin 空间是一个集合，其上配备其幂集的一个有界子格；众所周知，这些对象刻画了那些传递且完全有界的拟一致空间。我们把 Pervin 空间的无点概念称为 Frith frame，它由一个配备生成性有界子格的 frame 构成。本文引入并研究 Frith frame 的范畴，并证明拓扑空间与 frame 之间的经典对偶伴随可扩展为 Pervin 空间与 Frith frame 之间的对偶伴随。与 Pervin 空间的情形不同，传递且完全有界的拟一致 frame 范畴与 Frith frame 范畴之间不存在等价，但我们证明后者是前者的满全反射子范畴。我们还探讨了 Frith frame 从拟一致 frame 继承而来的完备性概念，给出了完备 Frith frame 的刻画，并描述了任意 Frith frame 的完备化。
英文摘要：
We lay down the foundations for a pointfree theory of Pervin spaces. A Pervin space is a set equipped with a bounded sublattice of its powerset, and it is known that these objects characterize those quasi-uniform spaces that are transitive and totally bounded. The pointfree notion of a Pervin space, which we call Frith frame, consists of a frame equipped with a generating bounded sublattice. In this paper we introduce and study the category of Frith frames and show that the classical dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames. Unlike what happens for Pervin spaces, we do not have an equivalence between the categories of transitive and totally bounded quasi-uniform frames and of Frith frames, but we show that the latter is a full coreflective subcategory of the former. We also explore the notion of completeness of Frith frames inherited from quasi-uniform frames, providing a characterization of those Frith frames that are complete and a description of the completion of an arbitrary Frith frame.
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