Let $\pi$ be a discrete group. A based topological space $X$ is called an Eilenberg-MacLane space of type $K(\pi,n)$ where $n\ge 1,$ if all the homotopy groups $\pi_k(X)$ are trivial except for $\pi_n(X),$ which is isomorphic to $\pi.$ Clearly, for such a space to exist when $n\ge 2,$ $\pi$ must be abelian.

Given any group $\pi,$ with $\pi$ abelian if $n\ge 2,$ there exists an Eilenberg-MacLane space of type $K(\pi,n).$ Moreover, this space can be constructed as a CW complex. It turns out that any two Eilenberg-MacLane spaces of type $K(\pi,n)$ are weakly homotopy equivalent. The Whitehead theorem then implies that there is a unique $K(\pi,n)$ space up to homotopy equivalence in the category of topological spaces of the homotopy type of a CW complex. We will henceforth restrict ourselves to this category. With a slight abuse of notation, we refer to any such space as $K(\pi,n).$ An important property of $K(\pi,n)$ is that, for $\pi$ abelian, there is a natural isomorphism $$ H^n(X;\pi) \isom [X,K(\pi,n)] $$ of contravariant set-valued functors, where $[X,K(\pi,n)]$ is the set of homotopy classes of based maps from $X$ to $K(\pi,n).$ Thus one says that the $K(\pi,n)$ are representing spaces for cohomology with coefficients in $\pi.$