Triangulations of monotone families I: Two-dimensional families

Let $K \subset {\mathbb R}^n$ be a compact definable set in an o-minimal structure over $\mathbb R$, e.g., a semi-algebraic or a subanalytic set. A definable family $\{ S_\delta|\> 0< \delta \in {\mathbb R} \}$ of compact subsets of $K$, is called a monotone family if $S_\delta \subset S_\eta$ for all sufficiently small $\delta > \eta >0$. The main result of the paper is that when $\dim K \le 2$ there exists a definable triangulation of $K$ such that for each (open) simplex $\Lambda$ of the triangulation and each small enough $\delta>0$, the intersection $S_\delta \cap \Lambda$ is equivalent to one of the five standard families in the standard simplex (the equivalence relation and a standard family will be formally defined). The set of standard families is in a natural bijective correspondence with the set of all five lex-monotone Boolean functions in two variables. As a consequence, we prove the two-dimensional case of the topological conjecture in [6] on approximation of definable sets by compact families. We introduce most technical tools and prove statements for compact sets $K$ of arbitrary dimensions, with the view towards extending the main result and proving the topological conjecture in the general case.