We study here several finiteness problems concerning affine Nash manifolds and Nash subsets . Three main results are: (i) A Nash function on a semialgebraic subset of has a Nash extension to an open semialgebraic neighborhood of in , (ii) A Nash set that has only normal crossings in can be covered by finitely many open semialgebraic sets equipped with Nash diffeomorphisms such that , (iii) Every affine Nash manifold with corners is a closed subset of an affine Nash manifold where the Nash closure of the boundary of has only normal crossings and can be covered with finitely many open semialgebraic sets such that each intersection for a Nash diffeomorphism .
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José F. Fernando, José Manuel Gamboa, Jesús M. Ruiz, Finiteness problems on Nash manifolds and Nash sets. J. Eur. Math. Soc. 16 (2014), no. 3, pp. 537–570DOI 10.4171/JEMS/439