Triangulations of Monotone Families I: Two-dimensional Families

Let K ⊂ Rn be a compact definable set in an o-minimal structure over R, e.g., a semi-algebraic or a subanalytic set. A definable family {Sδ| 0 < δ ∈ R} of compact subsets of K, is called a monotone family if Sδ ⊂ Sη for all sufficiently small δ > η > 0. The main result of the paper is that when dimK ≤ 2 there exists a definable triangulation of K such that for each (open) simplex Λ of the triangulation and each small enough δ > 0, the intersection Sδ ∩ Λ 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