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I am following the definitions of an o-minimal system as defined in "Tame Topology and O-minimal Structures" by Lou van den Dries.

It is immediate from the definition that the graph of $\sin(x)$ is not a tame set (intersect it with $y=0$). But what about a slightly rotated one? Or one which is both rotated and translated. To me they look to be tame (unless rotated by $\pi/4$). Is it correct that these sets are contained in some o-minimal system? And how can I easily 'recognize' tame sets? E.g. my intuition is that a collection of sets in $\mathbb{R}^2$ are tame if they do not invalidate the minimality axiom ($S_1$ contains exactly finite unions of points and open intervals). If so I can just complete with whatever sets needed in order for it to be a o-minimal structure.

And lastly, what about definable maps? I think that for semilinear sets the simplicial maps consititute a set of definable maps but there seems to be many more. How should I think of definable maps? (I am familiar with the monotonicity theorem).

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No, none of those translated or rotated graphs of $y=\sin x$ can be tame. For structure of definable sets, see a later chapter in that book. A finite union of cells of a certain simple kind. – GEdgar Sep 13 '12 at 14:17
@GEdgar: but is not possible to see that directly from the definition? – M.B. Sep 19 '12 at 12:24

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