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I apologize in advance if this question is too elementary for MO. I am new to the field of algebraic geometry.

I am dealing with a (real) algebraic variety $V$ of (Krull) dimension $n$. I keep reading that $V$ can be decomposed into differentiable manifolds of dimensions $0, 1, ..., n$.

While this seems plausible I could not find this statement in the literature. Is it true? I would also be glad about a reference.

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I believe this follows from results in Chapter 9 of Real Algebraic Geometry by Bochnak, Coste, and Roy. In particular, Proposition 9.1.8 implies the strata are Nash manifolds, satisfying certain "niceness" conditions.

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