Cohomological dimension of locally connected compacta

In this paper we investigate the cohomological dimension of cohomologically locally connected compacta with respect to principal ideal domains. We show the cohomological dimension version of the Borsuk–Siecklucki theorem: for every uncountable family {Kα}α∈A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α≠β such that dim(Kα∩Kβ)=n. As its consequences we shall investigate the equality of cohomological dimension and strong cohomological dimension and give a characterization of cohomological dimension by using a special base. Furthermore, we shall discuss the relation between cohomological dimension and dimension of cohomologically locally connected spaces.