Decomposing definable subsets of the plane
First some terminology: Let $X$ be a set and $Y_1, \dots, Y_l \subseteq X$, and put $\Y:= \{Y_1, \dots, Y_l\}$. We say that the $\Y$ partitions $X$ if $X = Y_1 \cup \cdots \cup Y_l$ and the $Y_j$-s are pairwise disjoint. Given $Z \subseteq X$, we say that $\Y$ is compatible with $Z$ if, for…