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Expansions which introduce no new open sets

Published online by Cambridge University Press:  12 March 2014

Gareth Boxall  and
Philipp Hieronymi
Gareth Boxall
Affiliation:
Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7600, South Africa, E-mail: gboxall@sun.ac.za
Philipp Hieronymi
Affiliation:
University of Illinois at Urbana-Champaign, Department of Mathematics, 1409 W. Green Street, Urbana, IL 61801, USA, E-mail: P@hieronymi.de
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Abstract

We consider the question of when an expansion of a first-order topological structure has the property that every open set definable in the expansion is definable in the original structure. This question has been investigated by Dolich, Miller and Steinhorn in the setting of ordered structures as part of their work on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give a further characterisation in the special case of an expansion by a generic predicate.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 111 - 121
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] Adler, H., A geometric introduction to forking and thorn-forking, Journal of Mathematical Logic, vol. 9 (2009), pp. 1–20.CrossRefGoogle Scholar
[2] Belegradek, O. and Zilber, B., The model theory of the field of reals with a subgroup of the unit circle, Journal of the London Mathematical Society. Second Series, vol. 78 (2008), pp. 563–579.CrossRefGoogle Scholar
[3] Berenstein, A., Dolich, A., and Onshuus, A., The independence property in generalized dense pairs of structures, this Journal, vol. 76 (2011), pp. 391–404.Google Scholar
[4] Berenstein, A., Ealy, C., and Günaydin, A., Thorn independence in the field of real numbers with a small multiplicative group, Annals of Pure and Applied Logic, vol. 150 (2007), pp. 1–18.CrossRefGoogle Scholar
[5] Berenstein, A. and Vassiliev, E., On lovely pairs of geometric structures, Annals of Pure and Applied Logic, vol. 161 (2010), pp. 866–878.CrossRefGoogle Scholar
[6] Chatzidakis, Z. and Pillay, A., Generic structures and simple theories, Annals of Pure and Applied Logic, vol. 95 (1998), pp. 71–92.CrossRefGoogle Scholar
[7] Dolich, A., Miller, C., and Steinhorn, C., Generic expansions of ordered structures, Preprint., 2009.Google Scholar
[8] Dolich, A., Structures having o-minimal open core, Transactions of the American Mathematical Society, vol. 362 (2010), pp. 1371–1411.CrossRefGoogle Scholar
[9] van den Dries, L., The field of reals with a predicate for the powers of two, Manuscripta Mathematica, vol. 54 (1985), pp. 187–195.CrossRefGoogle Scholar
[10] 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, Stability in model theory, II (Trento, 1987).CrossRefGoogle Scholar
[11] van den Dries, L., Densepairs of o-minimal structures, Fundamenta Mathematicae, vol. 157 (1998), pp. 61–78.CrossRefGoogle Scholar
[12] van den Dries, L. and Günaydin, A., The fields of real and complex numbers with a small multiplicative group, Proceedings of the London Mathematical Society, vol. 93 (2006), pp. 43–81.CrossRefGoogle Scholar
[13] Ealy, C. and Onshuus, A., Characterizing rosy theories, this Journal, vol. 72 (2007), pp. 919–940.Google Scholar
[14] Flum, J. and Ziegler, M., Topological model theory, Lecture Notes in Mathematics, vol. 769, Springer-Verlag, Berlin, 1980.CrossRefGoogle Scholar
[15] Günaydin, A., Model theory of fields with multiplicative groups, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2008.Google Scholar
[16] Günaydin, A. and Hieronymi, P., The real field with the rational points of an elliptic curve, Fundamenta Mathematicae, vol. 211 (2011), pp. 15–40.CrossRefGoogle Scholar
[17] Hrushovski, E. and Pillay, A., Groups definable in local fields and pseudo-finite fields, Israel Journal of Mathematics, vol. 85 (1994), pp. 203–262.CrossRefGoogle Scholar
[18] Miller, C., Tameness in expansions of the real field, Logic Colloquium '01, Lecture Notes in Logic, vol. 20, Association for Symbolic Logic, Urbana, IL, 2005, pp. 281–316.CrossRefGoogle Scholar
[19] Miller, C. and Speissegger, P., Expansions of the real line by open sets: o-minimality and open cores, Fundamenta Mathematicae, vol. 162 (1999), pp. 193–208.Google Scholar
[20] Onshuus, A., Properties and consequences of thorn-independence, this Journal, vol. 71 (2006), pp. 1–21.Google Scholar
[21] Pillay, A., First order topological structures and theories, this Journal, vol. 52 (1987), pp. 763–778.Google Scholar
[22] Zilber, B., Complex roots of unity on the real plane, Unpublished, 2003.Google Scholar