Abstract

Let $K$ be either a CW or a metric simplicial complex. We find sufficient conditions for the subspace inequality $$A\subset X, \quad K\in \text{\rm AE}(X)\Rightarrow K\in \text{\rm AE}(A).$$ 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 $$A\subset X\Rightarrow\dim_GA\leq\dim_GB.$$ for a certain class of paracompact spaces $X$ and an arbitrary abelian group $G$. As for the addition theorems $$\gather K\in \text{\rm AE}(A), \;\; L\in\text{\rm AE}(B)\Rightarrow K\ast L\in\text{\rm AE}(A\cup B),\\ \dim_G(A\cup B)\leq\dim_GA+\dim_GB+1, \endgather$$ we extend Dydak's theorems for metrizable spaces ($G$ is a ring with unity) to some classes of paracompact spaces.

Keywords: Dimension; cohomological dimension; absolute extensor; CW-complex; metric simplicial complex; subspace theorem; addition theorem.

MSC: 55M11, 54F45

Pages:  281$-$305     

Volume  61 ,  Issue  4 ,  2009