### 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_{ℤ}^{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 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.

Original language | English |
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Pages (from-to) | 213-222 |

Number of pages | 10 |

Journal | Fundamenta Mathematicae |

Volume | 171 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2002 Jan 1 |

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### Keywords

- ANR
- Cohomological dimension
- Cohomology locally n-connected compacta
- Descending chain condition

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Fundamenta Mathematicae*,

*171*(3), 213-222. https://doi.org/10.4064/fm171-3-2