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

Chris Miller; Patrick Speissegger

Fundamenta Mathematicae (1999)

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

## Access Full Article

top## Abstract

top## How to cite

topMiller, 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

top- [CP] Z. Chatzidakis and A. Pillay, Generic structures and simple theories, Ann. Pure Appl. Logic 95 (1998), 71-92.
- [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
- [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.
- [D3] L. van den Dries, Dense pairs of o-minimal structures, Fund. Math. 157 (1998), 61-78. Zbl0906.03036
- [DM] L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540. Zbl0889.03025

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.