Mathematics – General Topology
Scientific paper
2002-10-28
Topology and its Applications 140 (2004), 227-243
Mathematics
General Topology
17 pages, to appear in Topology and its Applications
Scientific paper
We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper: \proclaim{Theorem} Suppose X is a paracompact space. There is a CW complex K such that {a.} K is an absolute extensor of X up to homotopy, {b.} If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy. proclaim The proof is based on the following simple result (see 1.6). \proclaim{Theorem} Suppose X be a paracompact space and $f:A\to Y$ is a map from a closed subset A of X to a space Y. f extends over X if Y is the union of a family $\{Y_s\}_{s\in S}$ of its subspaces with the following properties: {a.} Each $Y_s$ is an absolute extensor of X, {b.} For any two elements s and t of S there is $u\in S$ such that $Y_s\cup Y_t\subset Y_u$, {c.} $A=\bigcup\limits_{s\in S} \int_A(f^{-1}(Y_s))$. proclaim That result implies a few well-known theorems of classical theory of retracts which makes it of interest in its own.
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