Geodesic
On definably proper maps
Mário J. Edmundo 1   ; Marcello Mamino 2   ; Luca Prelli 3   ;
1 Universidade Aberta Rua Braamcamp 90 1250-052 Lisboa, Portugal and CMAF Universidade de Lisboa Av. Prof. Gama Pinto 2 1649-003 Lisboa, Portugal
2 Laboratoire d'Informatique de l'École Polytechnique (LIX) Bâtiment Turing, bureau 2011 1 rue Honoré d'Estienne d'Orves Campus de l'École Polytechnique 91120 Palaiseau, France
3 CMAF Universidade de Lisboa Av. Prof. Gama Pinto 2 1649-003 Lisboa, Portugal
Fundamenta Mathematicae, Tome 233 (2016) no. 1, pp. 1-36 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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In this paper we work in o-minimal structures with definable Skolem functions, and show that: (i) a Hausdorff definably compact definable space is definably normal; (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 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 (and definably compact) in elementary extensions and o-minimal expansions.
DOI : 10.4064/fm96-12-2015
Keywords: paper work o minimal structures definable skolem functions hausdorff definably compact definable space definably normal continuous definable map between hausdorff locally definably compact definable spaces definably proper only proper morphism category definable spaces several other characterizations definably proper including involving existence limits definable types prove basic properties definably proper maps invariance definably proper definably compact elementary extensions o minimal expansions

Mário J. Edmundo  1   ; Marcello Mamino  2   ; Luca Prelli  3   ;  

1 Universidade Aberta Rua Braamcamp 90 1250-052 Lisboa, Portugal and CMAF Universidade de Lisboa Av. Prof. Gama Pinto 2 1649-003 Lisboa, Portugal
2 Laboratoire d'Informatique de l'École Polytechnique (LIX) Bâtiment Turing, bureau 2011 1 rue Honoré d'Estienne d'Orves Campus de l'École Polytechnique 91120 Palaiseau, France
3 CMAF Universidade de Lisboa Av. Prof. Gama Pinto 2 1649-003 Lisboa, Portugal 
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Mário J. Edmundo; Marcello Mamino; Luca Prelli;  . On definably proper maps. Fundamenta Mathematicae, Tome 233 (2016) no. 1, pp. 1-36. doi: 10.4064/fm96-12-2015

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