Triangulations of monotone families I: two-dimensional families

Let K⊂Rn be a compact definable set in an o-minimal structure over R, for example, a semi-algebraic or a subanalytic set. A definable family {Sδ|0η>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 Gabrielov and Vorobjov [‘Approximation of definable sets by compact families, and upper bounds on homotopy and homology’, J. London Math. Soc. (2) 80 (2009) 35–54] 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.