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## Fundamenta Mathematicae

2002 | 171 | 3 | 213-222
Tytuł artykułu

### Borsuk-Sieklucki theorem in cohomological dimension theory

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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.
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Tom
Numer
Strony
213-222
Opis fizyczny
Daty
wydano
2002
Twórcy
autor
• Instituto de Matemáticas, UNAM, Av. Universidad S/N, Col. Lomas de Chamilpa, 62210 Cuernavaca, Morelos, México
autor
• Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.
autor
• Instituto de Matemáticas, UNAM, Av. Universidad S/N, Col. Lomas de Chamilpa, 62210 Cuernavaca, Morelos, México
autor
• Division of Mathematical Sciences, Osaka Kyoiku University, Kashiwara, Osaka 582-8582, Japan
autor
• Steklov Institute of Mathematics, Gubkina 8, 117966 Moscow GSP-1, Russia
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