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Let    K  ⊂    R   nbe a compact definable set in an o‐minimal structure over   R  , for example, a semi‐algebraic or a subanalytic set. A definable family    {   S  δ|    0  <  δ  ∈  R  }    of compact subsets of   K   is called a  monotone family  ifS  δ   ⊂   S  ηfor all sufficiently small    δ  >  η  >  0   . The main result of the paper is that when    dim  K  ⩽  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 intersectionS  δ   ∩  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.

Triangulations of monotone families I: two‐dimensional families