Let $X$ be a manifold and consider the universal covering $$p:\tilde{X}\longrightarrow X$$ I know that this has a structure as a principal $G(\tilde{X})$-bundle but I can't manage to correctly define trivializations:
Let $U\subset X$ be a neighbourhood such that $p^{-1}(U)$ is a disjoint union of open sets each of which is mapped homemorphically into $U$ by (the restriction of) $p$. Thus the trivialization has to be something like $$\varphi:\bigcup_{\alpha\in\Sigma}U_\alpha\longrightarrow G(\tilde{X})\times U$$ which (I think) must have the form $$x\in U_\alpha\longmapsto (f_\alpha,p(x))$$ so the first component depends only on the $U_\alpha$ and the second is always the projection of $x$. My problem is that I don't see how to define this $f_\alpha$, it feels that there is no canonical way of doing it, and I have to make a choice, but I don't see how.
Just pick one of the $U_\alpha$; let's say it's $U_1$. The group $G(\tilde{X})$ acts simply and transitively on the collection of the $U_\alpha$, so for each $\alpha$ there is a unique $g\in G(\tilde{X})$ such that $g(U_1)=U_\alpha$. You then define $f_\alpha$ to be this $g$.
In other words, once you fix the value of $f_\alpha$ for one particular $\alpha$ however you want to, all the others are uniquely determined by the action of $G(\tilde{X})$.