Abstract.
We give a geometric proof of the following well-established theorem for o-minimal expansions of the real field: the Hausdorff limits of a compact, definable family of sets are definable. While previous proofs of this fact relied on the model-theoretic compactness theorem, our proof explicitly describes the family of all Hausdorff limits in terms of the original family.
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Lion, J.M., Speissegger, P. A geometric proof of the definability of Hausdorff limits. Sel. math., New ser. 10, 377 (2004). https://doi.org/10.1007/s00029-004-0360-z
DOI: https://doi.org/10.1007/s00029-004-0360-z