Co-Noetherian spaces
Xiangrui Li, Qingguo Li
TL;DR
The paper introduces co-Noetherian spaces to refine the taxonomy of $T_0$ spaces in non-Hausdorff topology, defining them via a finitary coverage property on open sets and establishing their core properties. It shows that co-Noetherian spaces are strong $R$-spaces and, in the $T_0$ case, sober, linking them to open well-filteredness under Noetherian assumptions and to a dual viewpoint on compactness in the strong topology. A categorical equivalence is proved between the category of $T_0$ co-Noetherian spaces and a subcategory of dcpo's, while investigating how Hoare and Smyth powerspaces interact with co-Noetherianity, including several counterexamples. The work also situates co-Noetherian spaces within KC-spaces and strong $R$-spaces, discusses product behavior, and presents a classification diagram with open questions guiding future research in non-Hausdorff topology and domain theory.
Abstract
In non-Hausdorff topology, many spaces exhibit significant separation properties, such as sober spaces, well-filtered spaces and d-spaces. These properties serve to fundamentally classify T0 topological spaces. In this paper, we introduce and study a new class of topological spaces called co-Noetherian spaces, which can refine the classification of T0 spaces. We discuss some basic properties of co-Noetherian spaces and obtain an equivalent characterization of compactness under the strong topology. Additionally, we investigate the connections among KC-spaces, strong R-spaces and co-Noetherian spaces. Moreover, we establish an equivalence between the category of T0 co-Noetherian spaces with continuous mappings and a subcategory of the poset category. Finally, we provide counterexamples to show that the Hoare powerspace of a T0 space may fail to be co-Noetherian, and that the Smyth powerspace of a co-Noetherian space need not be co-Noetherian.