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Extension theory of separable metrizable spaces with applications to dimension theory

Authors: Alexander Dranishnikov and Jerzy Dydak
Journal: Trans. Amer. Math. Soc. 353 (2001), 133-156
MSC (1991): Primary 55M10, 54F45
Published electronically: August 3, 2000
MathSciNet review: 1694287
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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 $K\wedge L\in AE(X\times Y).$

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Additional Information

Alexander Dranishnikov
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611

Jerzy Dydak
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996

Keywords: Dimension, cohomological dimension, ANR's, absolute extensors
Received by editor(s): July 14, 1995
Received by editor(s) in revised form: February 5, 1999
Published electronically: August 3, 2000
Additional Notes: The first and second authors were supported in part by grants DMS-9696238 and DMS-9704372, respectively, from the National Science Foundation.
Article copyright: © Copyright 2000 American Mathematical Society

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