I do not know whether there is a standard or some traditional ways to decide whether a semialgebraic set is connected or not.
One way I know is the cylindrical algebraic decomposition (CAD) algorithm. I have read some papers on the CAD algorithm, and they mentioned connectedness can be done by using CAD. However, I have not seen any explanation or example.
Can CAD do the connectedness of semialgebraic sets and how? Or any other theoretical way to decide the connectedness of semialgebraic sets?