FUNDAMENTA MATHEMATICAE 171 (2002) Borsuk–Sieklucki theorem in cohomological dimension theory by Margareta Boege (Cuernavaca), Jerzy Dydak (Knoxville, TN), Rolando Jim´ enez (Cuernavaca), Akira Koyama (Osaka) and Evgeny V. Shchepin (Moscow) Abstract. The Borsuk–Sieklucki theorem says that for every uncountable family {X α } α∈A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α = β such that dim(X α ∩ X β )= n. In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is clc n+1 Z , where n ≥ 1, and G is an Abelian group. Let {X α } α∈J be an uncountable family of closed subsets of X. If dim G X = dim G X α = n for all α ∈ J , then dim G (X α ∩ X β )= n for some α = β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski [C-K]. Independently, Dydak and Koyama [D-K] proved it for G being an arbitrary principal ideal domain and posed the question of validity of the Theorem for quasicyclic groups (see Problem 1 in [D-K]). As applications of the Theorem we investigate equality of cohomological dimension and strong cohomological dimension, and give a characterization of cohomological dimension in terms of a special base. 1. Introduction. Borsuk [Bo] and Sieklucki [S] investigated dimension properties of ANR-compacta and proved the following: 1.1. Theorem. Let {X α } α∈A be an uncountable family of n-dimensional closed subsets of an n-dimensional ANR-compactum. Then there exist α = β in A such that dim(X α ∩ X β )= n. 2000 Mathematics Subject Classification : 55M10, 54F45. Key words and phrases : cohomological dimension, cohomology locally n-connected compacta, ANR, descending chain condition. The second author is partially supported by grant DMS-0072356 from the National Science Foundation. The third author is partially supported by CONACYT (25314E). The fifth author is partially supported by Russian Foundation of Basic Research (99-01-00009) and CONACYT (980066). [213]