Extension theory of separable metrizable spaces with applications to dimension theory

Abstract:

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 $K\wedge L\in AE(X\times Y).$