What's the classifying space $B\mathbb{R}$ of the additive group of real numbers provided with the Euclidean topology ?
By the extension $\mathbb{Z} \hookrightarrow \mathbb{R} \twoheadrightarrow S^1$ there is a fibration $S^1 \to B\mathbb{R} \to \mathbb{C}P^\infty$. But I wonder, if there is an explicit description of $B\mathbb{R}$ ? Has this space a particular name ?
If we can construct a space, which we will call $EG$ such that our group acts transitively and freely, then we may set $BG=EG/G$. But if we have $\mathbb{R}$ act on $\mathbb{ER}=\mathbb{R}^2$ by $r∗(x,y)=(x+r,y)$, this will do the trick. Thus we get that $B\mathbb{R}=\mathbb{R}$. Their are other models as well.