Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings

In this paper we consider families of complex or real algebraic varieties. We prove that for almost all values of the parameters both the topology of the variety and its position in space will be the same. The set of singular values of the parameters is calculated constructively. In this paper we also isolate a class of families of polynomial mappings. For such families we prove the topological equivalence of almost all the mappings included in them. These results are applied to a proof of Zariski's theorem on the fundamental group of the complement to an algebraic hypersurface.