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Topological properties of sets definable in weakly o-minimal structures

Published online by Cambridge University Press:  12 March 2014

Roman Wencel
Roman Wencel*
Affiliation:
Mathematical Institute, University of WrocławPl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. E-mail: rwenc@math.uni.wroc.pl
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Abstract

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets of Cartesian products of definable sets, showing that if X, Y and S are non-empty definable sets and S is a large subset of X × Y. then for a large set of tuples , where k = dim(Y), the union of fibers is large in Y. Finally, given a weakly o-minimal structure , we find various conditions equivalent to the fact that the topological dimension in enjoys the addition property.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 841 - 867
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1]Arefiev, R., On monotonicity for weakly o-minimal structures, Algebra and model theory II (Pinus, A. G. and Ponomarev, A. N., editors), Novosibirsk, 1997, pp. 8–15.Google Scholar
[2]Macpherson, D., Marker, D., and Steinhorn, C., Weakly o-minimal structures and real closed fields, Transactions of the American Mathematical Society, vol. 352 (2000), pp. 5435–5483.CrossRefGoogle Scholar
[3]Pillay, A., On groups and fields definable in o-minimal structures, Journal of Pure and Applied Algebra, vol. 53 (1988), pp. 239–255.CrossRefGoogle Scholar
[4]van den Dries, L., Dimension of definable sets, algebraic boundedness and henselian fields, Annals of Pure and Applied Logic, vol. 45 (1989), pp. 189–209.CrossRefGoogle Scholar
[5]van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Notes Series, vol. 248, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar
[6]Verbovskiy, V.,Non-uniformly weakly o-minimal group, Algebra and model theory, 3 (Erlogol, 2001), Novosibirsk State Technical University, Novosibirsk, 2001, pp. 136–145 and 161.Google Scholar