TITLE: Classifying spaces of a profinite groups  
QUESTION [11 upvotes]: 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 ?

REPLY [2 votes]: There's a lot of literature on classifying spaces of the $p$-adic integers. For instance, there are two 45-year old "computations" of the complex K-theory of the $p$-adic integers, resulting in two different answers. An explanation of why they are different, only found recently, is by considering a more refined notion of a classifying space, which is a uniform space well-defined up to uniform homotopy equivalence (see, for lack of a better account, subsection 1.4 here).