# Expansions of the real line by open sets: o-minimality and open cores

Fundamenta Mathematicae (1999)

• Volume: 162, Issue: 3, page 193-208
• ISSN: 0016-2736

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## Abstract

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The open core of a structure ℜ := (ℝ,<,...) is defined to be the reduct (in the sense of definability) of ℜ generated by all of its definable open sets. If the open core of ℜ is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of ℝ is finite or uncountable, or if ℜ defines addition and multiplication and every definable open subset of ℝ has finitely many connected components, then the open core of ℜ is o-minimal.

## How to cite

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Miller, Chris, and Speissegger, Patrick. "Expansions of the real line by open sets: o-minimality and open cores." Fundamenta Mathematicae 162.3 (1999): 193-208. <http://eudml.org/doc/212420>.

@article{Miller1999,
abstract = {The open core of a structure ℜ := (ℝ,<,...) is defined to be the reduct (in the sense of definability) of ℜ generated by all of its definable open sets. If the open core of ℜ is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of ℝ is finite or uncountable, or if ℜ defines addition and multiplication and every definable open subset of ℝ has finitely many connected components, then the open core of ℜ is o-minimal.},
author = {Miller, Chris, Speissegger, Patrick},
journal = {Fundamenta Mathematicae},
keywords = {topology of the real line; o-minimality; open core; definable open sets; topological closure; definable set},
language = {eng},
number = {3},
pages = {193-208},
title = {Expansions of the real line by open sets: o-minimality and open cores},
url = {http://eudml.org/doc/212420},
volume = {162},
year = {1999},
}

TY - JOUR
AU - Miller, Chris
AU - Speissegger, Patrick
TI - Expansions of the real line by open sets: o-minimality and open cores
JO - Fundamenta Mathematicae
PY - 1999
VL - 162
IS - 3
SP - 193
EP - 208
AB - The open core of a structure ℜ := (ℝ,<,...) is defined to be the reduct (in the sense of definability) of ℜ generated by all of its definable open sets. If the open core of ℜ is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of ℝ is finite or uncountable, or if ℜ defines addition and multiplication and every definable open subset of ℝ has finitely many connected components, then the open core of ℜ is o-minimal.
LA - eng
KW - topology of the real line; o-minimality; open core; definable open sets; topological closure; definable set
UR - http://eudml.org/doc/212420
ER -

## References

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1. [CP] Z. Chatzidakis and A. Pillay, Generic structures and simple theories, Ann. Pure Appl. Logic 95 (1998), 71-92.
2. [D1] L. van den Dries, The field of reals with a predicate for the powers of two, Manuscripta Math. 54 (1985), 187-195. Zbl0631.03020
3. [D2] L. van den Dries, o-Minimal structures, in: Logic: From Foundations to Applications, Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, 137-185.
4. [D3] L. van den Dries, Dense pairs of o-minimal structures, Fund. Math. 157 (1998), 61-78. Zbl0906.03036
5. [DM] L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540. Zbl0889.03025

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