This is very much the same post as I posted at [math.stackexchange](https://math.stackexchange.com/questions/195203/some-intuition-behind-o-minimal-systems). 


I am following the definitions of an o-minimal system 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. 


I am familiar with the monotonicity theorem and how that may obstruct a cell decomposition of the aforementioned sets but I would really like to see from the very definition what goes wrong. It would be no problem if all lines in $\mathbb{R}^2$ were included but I cannot see how that is the case (all horizontal, vertical and $y=x$ are included by definition). 



Intuition behind o-minimal structures.
Let $B$ be a slight rotation of the graph of the sine function and lets say that $B \in S\_2$. Then one should be able to show from the definitions that the set $A$ of reals where $B$ has a "local maximum" is in $S\_1$. But the set $A$ is not a finite union of points and intervals.


**Edit 1:** Here are some details as to why $A$ has to be in $S\_1$.


First write a first order formula (using only the symbols $B$ and $\lt$ as binary predicates in addition to logical symbols) defining the elements of $A$. For instance, note that $a \in A$ if and only if:


$ \exists b,c,d [B(a,b) \wedge (c \lt a \lt d) \wedge \forall e,f [B(e,f) \wedge c \lt e \lt d \rightarrow f \leq b]] $.


Now the formula should guide you through the axioms you need to use to show that $A \in S\_1$. For example, you could start saying something like "$A$ is the projection onto the first coordinate of certain set, which is in $S\_4$ because...". Just remember that universal quantifiers correspond to complements of projections of complements and other logical symbols correspond to boolean operations.


**Edit 2:** On the other hand, if $B$ is the graph of the sine function rotated by $\pi/4$ and translated (say upwards) so that it doesn´t intersect the line $x=y$, then $B$ can actually be an element of an o-minimal structure (although not one that extends the ring structure of the reals). The reason is the following fact:



> 
> If $f,g: \mathbb{R} \to \mathbb{R}$
>  are two continuous increasing
>  bijections such that $x \lt f(x)$ and $x \lt g(x)$ for all $x \in \mathbb{R}$, then
>  the structures $(\mathbb{R},\lt,f)$
>  and $(\mathbb{R}, \lt, g)$ are
>  isomorphic.
> 
> 
> 


So taking $f$ as the function whose graph is $B$ and taking $g(x)=x+1$, we get that $(\mathbb{R},\lt,B)$ is o-minimal since $(\mathbb{R}, \lt, g)$ is o-minimal (being a "reduct" of the real field).


