  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.
