  We present an approach to cohomological dimension theory based on infinite
symmetric products and on the general theory of dimension called the extension
dimension. The notion of the extension dimension $\ExD(X)$ was introduced by
A.N.Dranishnikov \cite {D$_5$} in the context of compact spaces and CW
complexes.
  This paper investigates extension types of infinite symmetric products
$SP(L)$. One of the main ideas of the paper is to treat $\ExD(X)\leq SP(L)$ as
the fundamental concept of cohomological dimension theory instead of
$\dim_G(X)\leq n$. In a subsequent paper \cite{Dy$_6$} we show how properties
of infinite symmetric products lead naturally to a calculus of graded groups
which implies most of classical results of the cohomological dimension. The
basic notion in \cite{Dy$_6$} is that of homological dimension of a graded
group which allows for simultanous treatment of cohomological dimension of
compacta and extension properties of CW complexes. We introduce cohomology of
$X$ with respect to $L$ (defined as homotopy groups of the function space
$SP(L)^X$).
  As an application of our results we characterize all countable groups $G$ so
that the Moore space $M(G,n)$ is of the same extension type as the
Eilenberg-MacLane space $K(G,n)$. Another application is characterization of
infinite symmetric products of the same extension type as a compact (or
finite-dimensional and countable) CW complex.
