2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61394We 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 $1\to π_1(G)\to \tilde{G}\stackrel{p}\to G\to 1$ in the category of strictly properly $\bigvee $-definable groups with strictly properly $\bigvee $-definable homomorphisms, where $π_1(G)$ is the o-minimal fundamental group of $G$.69 pages, submitedLogicAlgebraic Topologytext