Submission date: 26 April 2014.

Abstract:

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 characterizations of definably proper including one involving the existence of limits of definable types, we prove the basic properties of definably proper maps and we prove the invariance of definably proper in elementary extensions and o-minimal expansions. We show also that these results hold for the notion of proper morphism in the category of o-minimal spectral spaces in analogy to what happens in real algebraic geometry (and also in algebraic geometry).

Mathematics Subject Classification: 03C64, 55N30

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Full text arXiv 1404.6634: pdf, ps.