A topological space with a single nonzero homotopy group. Use in conjunction with (algebraic-topology), (homotopy-theory), (homology-cohomology) or (classifying-spaces).
Given an integer $n \ge 1$ and a group $A$ (which has to be abelian if $n \ge 2$), the Eilenberg–MacLane space $K(A,n)$ (sometimes also denoted $B^nA$ to emphasize that it is an $n$-fold delooping of the group $A$) is a path-connected topological space whose homotopy groups satisfy: $$\pi_k\bigl( K(A,n) \bigr) = \begin{cases} A, & k = n; \\ 0, & k \neq n. \end{cases}$$ Such a space always exists, and it is unique up to homotopy.
A fundamental property of these spaces is that they classify cohomology: given a CW-complex $X$, the set of homotopy classes of maps $X \to K(A,n)$ is naturally in bijection with the $n$-th cohomology group of $X$ with coefficients in $A$: $$H^n(X;A) \cong [X, K(A,n)].$$ These spaces also occur as the fiber of the maps in a Postnikov tower.
