1,428 research outputs found
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.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1401.084
Integration and Cell Decomposition in -minimal Structures
We show that the class of -constructible functions is closed
under integration for any -minimal expansion of a -adic field
. This generalizes results previously known for semi-algebraic
and sub-analytic structures. As part of the proof, we obtain a weak version of
cell decomposition and function preparation for -minimal structures, a
result which is independent of the existence of Skolem functions. %The result
is obtained from weak versions of cell decomposition and function preparation
which we prove for general -minimal structures. A direct corollary is that
Denef's results on the rationality of Poincar\'e series hold in any -minimal
expansion of a -adic field .Comment: 22 page
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