  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 $G$.
