Suppose $\pi : P\rightarrow M$ is a principal $G$-bundle. I want to make this into a covering map by changing the topology of $P$.
By local triviality we can find for each $x\in M$ an open $U\subset M$ containing $x$ such that there is a diffeomorphism $\phi: \pi^{-1}(U)\rightarrow U\times G$. We can choose the following topology on $U\times G$: the opens are just $V\times \{g\}$ where $V\subset U$ open and $g\in G$ and then take all possible unions and intersections as the topology. We also have to add $V\times \emptyset$ to make it into a topology. Then define the topology on $P$ by requiring that $\phi$ is diffeomorphism.
Then we have that $\pi^{-1}(U)=\cup_{g\in G}\phi^{-1}(U\times \{g\})$ and then we can say that $\pi : P\rightarrow M$ is a covering.
Question
I am not sure at all if this construction actually really works, it is just an idea. I hope someone can clarify some things.
Edit
Suppose that we have a flat connection on $P$, i.e. a vector sub bundle $H\subset TP$ such that $d\pi_p|_{H_p}:H_p\rightarrow T_{\pi(p)} M$ is an isomorphism for all $p\in P$. Flatness implies that this distribution is integrable, using Frobenius. Now I want to use this to make $P$ a covering space for $M$. Let $p\in P$ and $U\subset P$ open containing $p$ such that we have local coordinates $(\chi_1,...,\chi_k,...,\chi_n)$ where $k=\dim M$ and $n=\dim P$ and $$\left(\frac{\partial}{\partial \chi_1}|_q,...,\frac{\partial}{\partial \chi_k}|_q\right)$$ is a basis for $H_q$ for all $q\in U$. If we look at $\pi(U)\subset M$ (which is open) and then take the inverse image, we get $\pi^{-1}(\pi(U))=\cup_{g\in G}U\cdot g$. Each $U\cdot g$ is open and they are disjoint so we have almost a covering, but we also need that $\pi$ is a homeomorphism from each $U\cdot g$ to $\pi(U)$. From here I am stuck. Any ideas how to change the topology of $P$ and make it work?