  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.
