**A Real Algebraic Geometry Example** <br> 

Semialgebraic sets are very nice: they are closed under Boolean operations (obvious) and projections (not so obvious, but old result of Tarski-Seidenberg). Semianalytic sets are not so nice, because they're not closed under projections. 

What is one to do if one wants to study them nonetheless? Shift the focus to  *projections* of semianalytics instead, a.k.a. subanalytic sets. Those sets are closed under projection by construction, but all of a sudden, the Boolean algebra property is not so clear. But that's where Gabrielov's theorem of the complement comes in: the complement of a subanalytic set is again subanalytic. We now have a nice structure in which reside all the natural geometric operations we may want to do.