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.



Decomposition of real algebraic varieties into manifolds
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](https://en.wikipedia.org/wiki/Nash_functions#Nash_manifolds), satisfying certain "niceness" conditions.


