Betti numbers of semialgebraic and sub-Pfaffian sets

Let X be a subset in [-1, 1](n0) subset of R-n0 defined by the formula X = {x(0) \ Q(1)x(1) Q(2)x(2) ... Q(v)x(v) ((x(0), x(1), ...,x(v)) is an element of X-v)}, where Q(i) is an element of {There Exists, For All}, Q(i) not equal Q(i+1), x(i) is an element of [-1, 1](ni), and X-v may be either an open or a closed set in being the difference between a finite CW-complex and its subcomplex. An upper bound on each Betti number of X is expressed via a sum of Betti numbers of some sets defined by quantifier-free formulae involving X-v. In important particular cases of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae with polynomials and Pfaffian functions respectively, upper bounds on Betti numbers of X-v are well known. The results allow to extend the bounds to sets defined with quantifiers, in particular to sub-Pfaffian sets.