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We prove new upper bounds on homotopy and homology groups of o‐minimal sets in terms of their approximations by compact o‐minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by quantifier‐free formulae and obtain, for the first time, a singly exponential bound on Betti numbers of sub‐Pfaffian sets.

Approximation of definable sets by compact families, and upper bounds on homotopy and homology