2014
DOI: 10.1515/advgeom-2013-0028
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A short geometric proof that Hausdorff limits are definable in any o-minimal structure

Abstract: Abstract. The aim of this note is to give yet another proof of the following theorem: given an arbitrary o-minimal structure on the ordered field of real numbers R and any definable family A of definable nonempty compact subsets of R n , then the closure of A in the sense of the Hausdorff metric (or, equivalently, in the Vietoris topology) is a definable family. In particular, any limit in the sense of the Hausdorff metric of a convergent sequence of subsets of a definable family is definable in the same omini… Show more

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Cited by 5 publications
(3 citation statements)
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“…We believe that the following proposition, which is a direct consequence of the Main Theorem in [15], can be useful in several contexts. Proposition 5.2.…”
Section: Appendixmentioning
confidence: 99%
“…We believe that the following proposition, which is a direct consequence of the Main Theorem in [15], can be useful in several contexts. Proposition 5.2.…”
Section: Appendixmentioning
confidence: 99%
“…Our first result shows how the Bewley Kohlberg result extends to general-sum games. Though it stems directly from previous results of the literature ( [17] and [11]), it did not appear explicitly before.…”
Section: Model and Resultsmentioning
confidence: 83%
“…if A is a finite union of sets, each of these sets being defined as the conjunction of finitely many weak or strict polynomial inequalities. The following proposition is a direct consequence of the main theorem in [11] :…”
Section: Model and Resultsmentioning
confidence: 84%