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We characterize strong cohomological dimension of separable metric spaces in terms of extension of mappings. Using this characterization, we discuss the relation between strong cohomological dimension and (ordinal) cohomological dimension and give examples to clarify their gaps. We also show that $In{d}_{G}X=di{m}_{G}X$ if X is a separable metric ANR and G is a countable Abelian group. Hence $di{m}_{\mathbb{Z}}X=dimX$ for any separable metric ANR X.

The Borsuk-Sieklucki theorem says that for every uncountable family ${X_{\alpha }}_{\alpha \in A}$ of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that $dim\left(X_{\alpha }\cap X_{\beta }\right)=n$. In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is $cl{c}_{\mathbb{Z}}^{n+1}$, where n ≥ 1, and G is an Abelian group. Let ${X_{\alpha }}_{\alpha \in J}$ be an uncountable family of closed subsets of X. If $di{m}_{G}X=di{m}_G{X}_{\alpha }=n$ for all α ∈ J, then $di{m}_G\left(X_{\alpha }\cap X_{\beta }\right)=n$ for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski...