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 $A$, $A$ is (locally) a submanifold, and hence we can easily define the dimension of $A$ to be the largest dimension at points at which $A$ is a submanifold. It is not hard to see that a semialgebraic set lies inside an algebraic subvariety of the same dimension.