I'm trying to reconcile two results on the classification of principal bundles. First, we have $\mathrm{Prin}\_G(X)$ (the equivalence classes of $G$-bundles on $X$) is isomorphic to $H^1(X;G)$ (the first Cech cohomology group of X -- I'm taking $G$ to be abelian). Second, we have $\mathrm{Prin}\_G(X)$ is isomorphic to $[X,BG]$, the set of homotopy equivalences of maps from $X$ to $BG$, the classifying space of $G$. If we now take $G=\mathbb{R}$, the real line viewed as an additive group, the first result seems to say we can have non-trivial bundles, while the latter seems to contradict that (since we can take $BG$ to be a point). How do I reconcile these? Does one result not apply in this case?


Thanks in advance!



Classification of principal bundles
The cohomology group $H^1(X,\mathbb{R})$ is the Cech cohomology group of the sheaf of differentiable functions over $X$, since there exist partition of unity, $H^1(X,\mathbb{R})=0.$ 


[What is the right version of "partitions of unity implies vanishing sheaf cohomology"](https://mathoverflow.net/questions/11567/what-is-the-right-version-of-partitions-of-unity-implies-vanishing-sheaf-cohomo)


