[Paper Review] Co-Noetherian spaces

Xiangrui Li, Qingguo Li|arXiv (Cornell University)|Jan 16, 2026

Fuzzy and Soft Set Theory0 citations

TL;DR

The paper introduces co-Noetherian spaces, studies their basic properties, relations to KC-spaces and strong R-spaces, and establishes a category equivalence with a subcategory of DCPOs, plus counterexamples for Hoare/Smyth powerspaces.

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.

Motivation & Objective

• Refine the classification of T0 spaces by introducing co-Noetherian spaces as a dual notion to Noetherian spaces.
• Investigate basic properties and equivalent characterizations of co-Noetherian spaces.
• Characterize compactness under the strong topology for co-Noetherian spaces.
• Explore connections between co-Noetherian spaces and KC-spaces and strong R-spaces.
• Establish an equivalence between the category of T0 co-Noetherian spaces and a subcategory of DCPOs; provide counterexamples for Hoare/Smyth powerspaces.

Proposed method

• Define co-Noetherian spaces as spaces where every closed set is a compact element in the lattice of closed sets.
• Prove equivalent formulations involving open/saturated sets and finite subfamilies (Prop. 4.2).
• Show that co-Noetherian T0 spaces are strong R-spaces (Prop. 4.4) and deduce sobriety (Cor. 4.5).
• Characterize when the strong topology is compact (Thm 4.7) and relate this to Noetherian + co-Noetherian conditions (Thm 4.7, Cor. 4.9).
• Demonstrate that the Hoare and Smyth power constructions may fail to preserve co-Noetherianity (Exs. 4.17–4.18).
• Establish an equivalence between Co-NOE (category of T0 co-Noetherian spaces with continuous maps) and C-DCPO (category of controllable dcpos with upper-continuous maps) (Thm 4.22).

Experimental results

Research questions

• RQ1 What are the fundamental properties and equivalent characterizations of co-Noetherian spaces?
• RQ2 How do co-Noetherian spaces relate to KC-spaces and strong R-spaces?
• RQ3 Can the category of T0 co-Noetherian spaces be realized as a subcategory of DCPOs?
• RQ4 Do Hoare and Smyth power constructions preserve the co-Noetherian property?
• RQ5 Under what conditions is the strong topology compact for spaces in this class?

Key findings

• Co-Noetherian spaces are strong R-spaces, and hence strong R-space properties follow from the co-Noetherian condition.
• Every T0 co-Noetherian space is sober.
• The strong topology is compact exactly when the space is Noetherian and co-Noetherian (equivalently Noetherian plus well-filtered/open well-filtered under certain formulations).
• There is a categorical equivalence between the category of T0 co-Noetherian spaces and a subcategory of controllable dcpos (DCPOs) with upper-continuous maps (Co-NOE ≃ C-DCPO).
• Hoare powerspace of a T0 space and Smyth powerspace of a co-Noetherian space need not be co-Noetherian, providing important counterexamples to preservation under these constructions.