On definably proper maps

In this paper we work in o-minimal structures with definable Skolem functions and show that a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is proper morphism in the category of definable spaces. We give several other characterizations of definably proper including one involving the existence of limits of definable types. We also prove the basic properties of definably proper maps and the invariance of definably proper in elementary extensions and o-minimal expansions.