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.
@article{bwmeta1.element.bwnjournal-article-fmv162i3p193bwm,
author = {Chris Miller and Patrick Speissegger},
title = {Expansions of the real line by open sets: o-minimality and open cores},
journal = {Fundamenta Mathematicae},
volume = {159},
year = {1999},
pages = {193-208},
zbl = {0946.03045},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv162i3p193bwm}
}
Miller, Chris; Speissegger, Patrick. Expansions of the real line by open sets: o-minimality and open cores. Fundamenta Mathematicae, Tome 159 (1999) pp. 193-208. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv162i3p193bwm/
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