  In this paper we work in an arbitrary o-minimal structure with definable
Skolem functions and we prove that definably connected, locally definable
manifolds are uniformly definably path connected, have an admissible cover by
definably simply connected, open definable subsets and, definable paths and
definable homotopies on such locally definable manifolds can be lifted to
locally definable covering maps. These properties allows us to obtain the main
properties of the general o-minimal fundamental group, including: invariance
and comparison results; existence of universal locally definable covering maps;
monodromy equivalence for locally constant o-minimal sheaves - from which one
obtains, as in algebraic topology, classification results for locally definable
covering maps, o-minimal Hurewicz and Seifert - van Kampen theorems.
