As a topological group, a profinite group $G$ has a classifying space $BG$. On the other hand, since $G = \underleftarrow{\lim}\; G/U\;\;(U \le G$ open$)$ is an inverse limit of finite groups, we also have an inverse system $B(G/U)$ of topological spaces and can form the space $\underleftarrow{\lim}B(G/U)$.

**Question 1:** How do $BG$ and $\underleftarrow{\lim}B(G/U)$ relate ? Can we, for example say, that $\underleftarrow{\lim}B(G/U)$ is a classifying space for $G$ ?

**Question 2:** Are there futher results about classifying spaces of profinite groups ?