A short geometric proof that Hausdorff limits are definable in any o-minimal structure
-
Beata Kocel-Cynk
,
Wiesław Pawłucki
und
Anna Valette
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 ℝn, 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 o-minimal structure. The original proofs by Bröcker [1], Marker and Steinhorn [7], Pillay [11] (see also van den Dries [15]) were based on model theory. Lion and Speissegger [6] gave a geometric proof of the theorem. Our proof below is based on the idea of Lipschitz cell decompositions.
© 2014 by Walter de Gruyter GmbH & Co.
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Artikel in diesem Heft
- Masthead
- Real hypersurfaces of complex and quaternionic hyperbolic spaces
- Toric geometry of the 3-Kimura model for any tree
- Invariant surfaces in Sol3 with constant mean curvature and their computer graphics
- A short geometric proof that Hausdorff limits are definable in any o-minimal structure
- Seshadri constants on smooth threefolds
- Visible shoreline results for staircase paths in ℝd
- Characterization of isometric embeddings of Grassmann graphs
- Typical simplicially convex bodies
- On successive radii of p-sums of convex bodies
- Resolving sets for four families of distance-regular graphs
- White-black graphs and right-angled hyperbolic handlebodies
- Scalar and Ricci curvatures of special contact slant submanifolds in Sasakian space forms
- Non-factorial quartic double solids
- Collineation groups of topological parallelisms
