Let $U\subsetneq\mathbb{R}^n$ be an open semi-algebraic set which is not Zariski open (the case I have in mind is that of an open ball $B(0,1)$). A function $f:U\rightarrow \mathbb{R}$ is called
(1) Nash if $f$ is analytic and there exists some polynomial $P\in \mathbb{R}[x_1,...,x_{n+1}] $ s.t. $\forall x\in U:P(x,f(x))=0$. The set of all such functions will be denoted by $\mathcal{N}(U)$.
(2) A continuous rational function if it is continuous and there exists a rational function $g$ regular on some open subset $V\subseteq\mathbb{R}^n$ such that $f\restriction_{U\cap V}=g\restriction_{U\cap V}$. The set of all such functions will be denoted by $\mathcal{CR}(U)$.
(3) Arc-analytic if for every analytic arc $\gamma :(-\epsilon , \epsilon )\rightarrow U,$ $f\circ \gamma $ is analytic. The set of all such functions will be denoted by $\mathcal{AA}(U)$.
My questions are:
(i) With respect to what topologies does each of these classes of functions become a sheaf? If there is more than one such for each one, how do the stalks change if we choose a different one?
(ii) What can we say about the stalks of these sheaves at a point $x\in U$? (other than the obvious statement that there's an injection $\mathcal{N}_x\hookrightarrow \mathcal{AA}_x$ with respect to any topology that makes both of these classes sheaves)
