Covers of groups definable in o-minimal structures

We develop in this paper the theory of covers for Hausdorff properly $\bigvee$-definable manifolds with definable choice in an o-minimal structure $\N$. In particular, we show that given an $\N$-definably connected $\N$-definable group $G$ we have $1\to \pi_1(G)\to \tilde{G}\stackrel{p}\to G\to 1$ in the category of strictly properly $\bigvee$-definable groups with strictly properly $\bigvee$-definable homomorphisms, where $\pi_1(G)$ is the o-minimal fundamental group of $G$.