Cohomologies of countable unions of closed sets with applications to Cantor manifolds

Abstract

The main result: Let $X$ be a compact spacce, $A$ be its closed subset, and $X/A=\underset{i=1}{\overset{\infty}{\bigcup}}F_i$, where $F_i$ are closed subsets of $X/A$ such that dim$(F_i\cap F_j) \leq n-1$ for $i\neq j$. Then the natural homomorphism of $H^r(X,A;G)$ into the direct sum $\underset{i=1}{\overset{\infty}{\Pi}}H^r(A \cup F_i , A;G)$ is a monomorphism for $r \geq n+1$. Some applications of this result to strong Cantor manifolds (with respect to a group $G$) are obtained.