On definably proper maps

In this paper we work in o-minimal structures with definable Skolem func-tions, and show that: (i) a Hausdorff definably compact definable space is definably nor-mal; (ii) a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is a proper morphism in the category of dfinable 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 (and definably compact) in elementary extensions and o-minimal expansions.