2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/155719We 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.To appear in Fundamenta MathematicaeAlgebraic TopologyGeneral TopologyGeometric Topology54F45, 55M10, 55N99, 55Q40, 55P20text