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 $\{S_{\delta }\mathrm{\mid }0<\delta \in \mathbb{R}\}$ 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 $\mathrm{\Lambda }$ of the triangulation and each small enough $\delta >0$, the intersection $S_{\delta }\cap \mathrm{\Lambda }$ 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.