Extension dimension for paracompact spaces

We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper: m. Suppose X is a paracompact space. There is a CW complex K such that (a) n absolute extensor of X up to homotopy, W 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. roof is based on the following simple result (see Theorem 1.2). m. Let X be a paracompact space. Suppose a space Y is the union of a family {Ys}s∈S of its subspaces with the following properties: (a) s is an absolute extensor of X, y two elements s and t of S there is u∈S such that Ys∪Yt⊂Yu. A→Y is a map from a closed subset A to Y such that A=⋃s∈SIntA(f−1(Ys)), then f extends over X. esult implies a few well-known theorems of classical theory of retracts which makes it of interest in its own.