  In this paper we introduce the notion of Blow-semialgebraic triviality
consistent with a compatible filtration for an algebraic family of algebraic
sets, as an equisingularity for real algebraic singularities. Given an
algebraic family of 3-dimensional algebraic sets defined over a nonsingular
algebraic variety, we show that there is a finite subdivision of the parameter
algebraic set into connected Nash manifolds over which the family admits a
Blow-semialgebraic trivialisation consistent with a compatible filtration. We
show a similar result on finiteness also for a Nash family of 3-dimensional
Nash sets through the Artin-Mazur theorem. As a corollary of the arguments in
their proofs, we have a finiteness theorem on semialgebraic types of polynomial
mappings from the 2-dimensional Euclidean space to the p-diemnsional Euclidean
space.
