Abstract
Let K⊂Rn be a compact definable set in an ominimal structure over R, for example, a semialgebraic or a subanalytic set. A definable family {Sδ0<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 lexmonotone Boolean functions in two variables. As a consequence, we prove the twodimensional 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.
Original language  English 

Pages (fromto)  10131051 
Number of pages  42 
Journal  Proceedings of the London Mathematical Society 
Volume  111 
Issue number  5 
Early online date  11 Nov 2015 
DOIs  
Publication status  Published  11 Nov 2015 
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Nicolai Vorobjov
Person: Research & Teaching