Electronic Version 23 Spring (1998)

J. S. Choi and G. Kozlowski

We weaken the conditions of Sieklucki's Theorem by using cohomological local connectivity and cohomological dimension based on Alexander-Spanier cohomology with compact supports in a countable principal ideal domain L. We could also use ech cohomology. The theorem states that if X is a clcⁿ locally compact separable metric space with dim_LX = n and {X_\lambda }_{\lambda \in \Lambda} is an uncountable collection of closed subsets of X with dim_LX_\lambda = n for all \lambda, then there are two distinct indices \mu, \lambda \in \Lambda such that dim_L(X_\mu \cap X_\lambda) = n. The proof combines the fact the family of submodules of a finitely generated L-module is countable with a Mayer-Vietoris argument.

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