Nous définissons deux notions nouvelles en géométrie analytique réelle, celle de fonction Nash-analytique et celle de faisceau semi-cohérent. Avec ces notions, nous obtenons des théorèmes de cohérence analogues à ceux du cas complexe (théorème de cohérence d’Oka, théorème de l’image directe, cohérence d’un ensemble analytique complexe).

We study here several finiteness problems concerning affine Nash manifolds $M$ and Nash subsets $X$. Three main results are: (i) A Nash function on a semialgebraic subset $Z$ of $M$ has a Nash extension to an open semialgebraic neighborhood of $Z$ in $M$, (ii) A Nash set $X$ that has only normal crossings in $M$ can be covered by finitely many open semialgebraic sets $U$ equipped with Nash diffeomorphisms $(u_1,\cdots ,u_m):U\to {ℝ}^m$ such that $U\cap X=\{u_1\cdots u_r=0\}$, (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash manifold...