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In this paper we introduce the notion of Blow-semialgebraic trivialityconsistent with a compatible filtration for an algebraic family of algebraicsets, as an equisingularity for real algebraic singularities. Given analgebraic family of 3-dimensional algebraic sets defined over a nonsingularalgebraic variety, we show that there is a finite subdivision of the parameteralgebraic set into connected Nash manifolds over which the family admits aBlow-semialgebraic trivialisation consistent with a compatible filtration. Weshow a similar result on finiteness also for a Nash family of 3-dimensionalNash sets through the Artin-Mazur theorem. As a corollary of the arguments intheir proofs, we have a finiteness theorem on semialgebraic types of polynomialmappings from the 2-dimensional Euclidean space to the p-diemnsional Euclideanspace.

Finiteness theorem on Blow-semialgebraic triviality for a family of 3-dimensional algebraic sets