Example

Endowed with the mapping $p:\mathbf{R}\rightarrow \mathbf{S}^1,t\mapsto e^{2\pi it}$, and with the operation of $\mathbf{Z}$ by translation, the space $\mathbf{R}$ is a principal covering space with group $\mathbf{Z}$. The space $\mathbf{S}^1$ is thus a classifying space for the group $\mathbf{Z}$.

For every discrete group G, we shall construct in the following number a metrisable space which is a classifying space for G.