Similarly as algebraic subvarieties, finite union and intersection of semialgebraic sets is still a semialgebraic set. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski-Seidenberg theorem says that they are also closed under projection.
On a dense open subset of , is (locally) a submanifold, and hence we can easily define the dimension of to be the largest dimension at points at which is a submanifold. It is not hard to see that a semialgebraic set lies inside an algebraic subvariety of the same dimension.
- 1 Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42. http://www.ams.org/mathscinet-getitem?mr=89k:32011MR 89k:32011
|Date of creation||2013-03-22 16:46:10|
|Last modified on||2013-03-22 16:46:10|
|Last modified by||jirka (4157)|
|Defines||dimension of a semialgebraic set|