A (co)homology theory for some preordered topological spaces

TL;DR

This paper develops a (co)homology theory for topological spaces with a specialization preorder, improving previous methods by accounting for the space's decomposition into poset and non-poset parts.

Contribution

It introduces a novel (co)homology framework that considers the decomposition of spaces into poset and complementary parts, enhancing existing approaches.

Findings

Spaces decompose into poset and non-poset parts.

The new (co)homology accounts for this decomposition.

Improves understanding of topological spaces with preorders.

Abstract

The aim of this short note is to develop a (co)homology theory for topological spaces together with the specialisation preorder. A known way to construct such a (co)homology is to define a partial order on the topological space starting from the preorder, and then to consider some (co)homology for the obtained poset; however, we will prove that every topological space with the above preorder consists of two disjoint parts (one called 'poset part', and the other one called 'complementary part', which is not a poset in general): this suggests an improvement of the previous method that also takes into account the poset part, and this is indeed what we will study here.