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I have a strong suspicion that yes, but as I am not a specialist in o-minimal structures, I thought that I might have overlooked some corner case.

The precise statement is as follows: let $X \subset \mathbb{R}^n$ be a set definable in an o-minimal structure. Then there exists a point $P \in X$ and an open neighbourhood $O \subset X$ of $P$ such that $O$ is a closed analytic subset of a real analytic manifold embedded into some open $U \subset \mathbb{R}^n$

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  • $\begingroup$ If I recall correctly, it is known that for every constant $k$, such sets are locally $C^k$, but with analyticity this is either open or false. $\endgroup$ Mar 23 '16 at 13:45
  • $\begingroup$ Dear Emil, would it help if one assumes that the real field is expanded only by locally analytic functions? (because in practice typical o-minimal expansions of reals, like $R_{an}$ and $R_{exp}$, are such) $\endgroup$ Mar 23 '16 at 14:29
  • $\begingroup$ I don’t really know, but I would find it surprising. Even starting from nice (locally analytic) functions, one can define functions that are very wild. Now, in an o-minimal structure functions cannot get too wild, but this is a top-down argument quite orthogonal to how the functions were generated. Basically, I can imagine this to work only in presence of powerful quantifier elimination, but then the result would not really say anything interesting. $\endgroup$ Mar 23 '16 at 15:38
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    $\begingroup$ (What I was recalling in the first comment can be found in www1.maths.leeds.ac.uk/maloa/lecturenotes/lyon.pdf , Chapter 8, in particular Theorems 8.6 and 8.7.) $\endgroup$ Mar 23 '16 at 15:48
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No.

In their paper Quasianalytic Denjoy-Carleman classes and o-minimality, J. AMS, vol. 16 (4), 2003, p. 751—777, Rolin, Speissegger and Wilkie show that there exists a function on $[-1,1]$ which belongs to an o-minimal structure but is nowhere analytic.

Their theorem 1 (p. 752) asserts that for appropriate choices of a sequence $M=(M_n)$, the class $\mathcal C_M$ of $\mathcal C^\infty$ functions $f$ such that $\mathopen|f^{(n)}\mathclose|\leq A^{n+1} M_n$ (for some $A$) generates an o-minimal structure. The precise condition is that $(M_n)$ satisfies the two conditions:

  1. $\sum_{n=0}^\infty\frac{ M_n}{M_{n+1}}=+\infty$;
  2. $(M_n/n!)^2 \leq (M_{n-1}/(n-1)!) (M_{n+1}/(n+1)!)$.

By theorem 2 (2), p. 752, there exists such a sequence $(M_n)$ and a function $f$ in the corresponding class which is nowhere analytic. This furnishes an o-minimal structure without a real analytic cell-decomposition (Corollary, (2), p. 752).

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  • $\begingroup$ I guess the situation is still unknown if we replace the word "analytic" by "$C^\infty$"? $\endgroup$
    – Todd Trimble
    Mar 23 '16 at 17:22
  • $\begingroup$ @ToddTrimble: The answer is also no. See my last comment below the question. $\endgroup$ Mar 23 '16 at 17:23
  • $\begingroup$ @ACL: Do you happen to know anything about the follow-up question mentioned in the comments, that is, when the o-minimal structure is an expansion of the real field only by locally analytic functions? $\endgroup$ Mar 23 '16 at 17:25
  • $\begingroup$ @EmilJeřábek : I would guess that every function in this structure is piecewise locally analytic. $\endgroup$
    – ACL
    Mar 23 '16 at 17:29
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    $\begingroup$ @EmilJeřábek Ah, thank you. Just to go straight to the source: Olivier Le Gal and Jean-Philippe Rolin, An o-minimal structure which does not admit $C^\infty$ cellular decomposition, Ann. Inst. Fourier (Grenoble), 59 (2009), pp. 543–562. $\endgroup$
    – Todd Trimble
    Mar 23 '16 at 17:36

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