Extension theory of separable metrizable spaces with applications to dimension theory

The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results:

Generalized Eilenberg-Borsuk Theorem. Let $L$ be a countable CW complex. If $X$ is a separable metrizable space and $K\ast L$ is an absolute extensor of $X$ for some CW complex $K$, then for any map $f:A\to K$, $A$ closed in $X$, there is an extension $f':U\to K$ of $f$ over an open set $U$such that $L\in AE(X-U)$.

Theorem. Let $K,L$ be countable CW complexes. If $X$ is a separable metrizable space and $K\ast L$ is an absolute extensor of $X$, then there is a subset $Y$ of $X$ such that $K\in AE(Y)$ and $L\in AE(X-Y)$.

Theorem. Suppose $G_{i},\ldots ,G_{n}$ are countable, non-trivial, abelian groups and $k>0$. For any separable metrizable space $X$ of finite dimension $\dim X>0$, there is a closed subset $Y$ of $X$ with $\dim _{G_{i}} Y=\max (\dim _{G_{i}} X-k,1)$ for $i=1,\ldots ,n$.

Theorem. Suppose $W$ is a separable metrizable space of finite dimension and $Y$ is a compactum of finite dimension. Then, for any $k$, $0<k<\dim W-\dim Y$, there is a closed subset $T$ of $W$such that $\dim T=\dim W-k$ and $\dim (T\times Y)=\dim (W\times Y)-k$.

Theorem. Suppose $X$ is a metrizable space of finite dimension and $Y$ is a compactum of finite dimension. If $K\in AE(X)$ and $L\in AE(Y)$ are connected CW complexes, then $K\wedge L\in AE(X\times Y).$