Co-Noetherian spaces

TL;DR

This paper introduces co-Noetherian spaces, a new class of T0 topological spaces, explores their properties, characterizations, and relationships with existing spaces, and examines their behavior under certain power space constructions.

Contribution

It defines co-Noetherian spaces, studies their fundamental properties, characterizations, and categorical equivalences, and analyzes their behavior in power space constructions.

Findings

Co-Noetherian spaces form a new class refining T0 space classification.

An equivalent characterization of compactness in co-Noetherian spaces is provided.

Counterexamples show certain power spaces may not be co-Noetherian.

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.