Subspace and addition theorems for extension and cohomological dimensions. A problem of Kuzminov

Abstract

Let $K$ be either a CW or a metricsimplicial complex. We find sufficient conditions for the subspace inequality$$A\subset X,Kın\text{rm AE}(X)\Rightarrow Kın\text{rm AE}(A).$$For the Lebesgue dimension ($K=S^n$) our result is a slightgeneralization of Engelking's theorem for a strongly hereditarilynormal space $X$. As a corollary we get the inequality$$A\subset X\Rightarrow {\dim }_{G}A\le {\dim }_{G}B.$$for a certain class of paracompact spaces $X$ and an arbitrary abelian group $G$.As for the addition theorems$$\text{gatherK}ın\text{rm AE}(A),Lın\text{rm AE}(B)\Rightarrow K\ast Lın\text{rm AE}(A\cup B),{\dim }_G(A\cup B)\le {\dim }_{G}A+{\dim }_{G}B+1,\text{endgather}$$we extend Dydak's theorems for metrizable spaces ($G$ is a ring with unity) to some classes of paracompact spaces.