Andrei Gabrielov Purdue University Let S(t), be a one-parametric decreasing family of compact sets in a compact subset K of R^n. The family S(t) and the compact K are assumed to be "tame" (for example, real semialgebraic). Problem: To construct a triangulation of K such that the intersection of S(t) with each open simplex is homeomorphic to one of the finitely many "standard" combinatorial types. The conjectural answer to this problem will be given, and related results concerning geometry and topology of real semialgebraic sets (and more general "tame" sets) will be presented.