Can we write every semi-algebraic set $A \subset \mathbb{R}^m$ as the disjoint union of finitely many Nash submanifolds of $\mathbb{R}^m$? By Nash submanifolds are meant those manifolds that are smooth and semi-algebraic.
2026-03-25 18:59:08.1774465148
Can every semi-algebraic set be written as a finite union of disjoint Nash submanifolds?
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Yes, a semi-algebraic cell decomposition does this already. Any of the usual texts on semi-algebraic geometry will prove this - for instance, Bochnak-Coste-Roy's Real Algebraic Geometry or Lou van den Dries' Tame Topology and O-Minimal Structures definitely have this, and both Michel Coste's notes on o-minimal geometry or semi-algebraic geometry also contain proofs.