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Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta BoegeJerzy DydakRolando JiménezAkira KoyamaEvgeny V. Shchepin — 2002

Fundamenta Mathematicae

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 d i m ( X α X β ) = n . In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is c l c n + 1 , where n ≥ 1, and G is an Abelian group. Let X α α J be an uncountable family of closed subsets of X. If d i m G X = d i m G X α = n for all α ∈ J, then d i m 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...

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