Download PDF - Expansions of the real line by open sets: o-minimality and open coresAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

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

Authors 1, 2, 2, 3

Affiliations

  1. The Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1
  2. Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, OH 43210, U.S.A.
  3. Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, U.S.A.

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.

Bibliography

  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.
  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.
  5. [DM] L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540.
Fundamenta Mathematicae
1999/162/3
Export to PBN, Opens in new tab
Pages:
193-208
Main language of publication
English
Received
1998-03-23
Accepted
1999-07-22
Published
1999
Exact and natural sciences