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 $X/A=\underset{i=1}{\bigcup ^{\mathrm{\infty }}}{F}_i$, where $F_i$ are closed subsets of $X/A$ such that dim$(F_i\cap F_j)\le n-1$ for $i\ne j$. Then the natural homomorphism of $H^r(X,A;G)$ into the direct sum $\underset{i=1}{\stackrel{\mathrm{\infty }}{\mathrm{\Pi }}}{H}^r(A\cup F_i,A;G)$ is a monomorphism for $r\ge n+1$. Some applications of this result to strong Cantor manifolds (with respect to a group $G$) are obtained.