In this paper we give a new bound on the sum of the Betti numbers of closed semi-algebraic sets. This extends a well-known bound due to Oleinik and Petrovsky [28], Thom [35] and Milnor [27]. In separate papers they proved that the sum of the Betti numbers of a semi-algebraic set S ae R k; defined by P 1 0; : : : ; P s 0; deg(P i) d; 1 i s; is
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