We study upper bounds on topological complexity of sets definable in o-minimal structures over the reals, for example semi-algebraic or subanalytic sets.
We suggest a construction for approximating sets in a large class, including sets defined by arbitrary Boolean combinations of equations and inequalities, by compact sets.
The homotopies and homologies of these compact sets bound from above the homotopies and homologies of the approximated sets.
We conjecture that for approximations of high enough accuracy, the approximated sets and the corresponding compact sets are homotopy equivalent. This conjecture is proved for two-dimensional sets in R^n.
Joint work with A. Gabrielov.
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