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Consider an $C^\infty$ abstract stratification $A$ (in the Thom-Mather sense, see Mather's note).


Can we embed $A$ in some $\mathbb{R}^n$ (or in an analytic manifold) as a subanalytic set?

If not, what could be the problem?


This apparently was shown by Laurent Noirel in his thesis for the $C^k$ case, $1 \leq k < \infty$.

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