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

V. V.,Fedorchuk

Let $K$ be either a CW or a metric simplicial complex. We find sufficient conditions for the subspace inequality $$Aubset X, \quad Kın ext{\rm AE}(X)\Rightarrow Kın ext{\rm AE}(A).$$ For the Lebesgue dimension ($K=S^n$) our result is a slight generalization of Engelking's theorem for a strongly hereditarily normal space $X$. As a corollary we get the inequality $$Aubset X\Rightarrow\dim_GAłeq\dim_GB.$$ for a certain class of paracompact spaces $X$ and an arbitrary abelian group $G$. As for the addition theorems $$\gather Kın ext{\rm AE}(A), \;\; Lınext{\rm AE}(B)\Rightarrow Kst Lınext{\rm AE}(A\cup B),\\ \dim_G(A\cup B)łeq\dim_GA+\dim_GB+1, \endgather$$ we extend Dydak's theorems for metrizable spaces ($G$ is a ring with unity) to some classes of paracompact spaces.