For any Abelian group 
and any natural number 
,
there is a unique space (up to homotopy
type) such that all homotopy groups except for
the 
th
are trivial (including the 0th homotopy groups,
meaning the space is pathwise-connected),
and the 
th
homotopy group is isomorphic
to the group 
. In the case where 
, the group 
can be non-Abelian as well.
Eilenberg-Mac Lane spaces have many important applications. One of them is that every topological space has the homotopy type of an iterated fibration of Eilenberg-Mac Lane spaces (called a Postnikov system). In addition, there is a spectral sequence relating the cohomology of Eilenberg-Mac Lane spaces to the homotopy groups of spheres.
