A compact semialgebraic set admits a semialgebraic triangulation such that the family of open simplexes forms a Whitney stratification and is compatible with a finite number of given semialgebraic subsets.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-20, author = {Masahiro Shiota}, title = {Whitney triangulations of semialgebraic sets}, journal = {Annales Polonici Mathematici}, volume = {85}, year = {2005}, pages = {237-246}, zbl = {1091.14016}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-20} }
Masahiro Shiota. Whitney triangulations of semialgebraic sets. Annales Polonici Mathematici, Tome 85 (2005) pp. 237-246. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-20/