We first define the topology of semi-algebraic sets and study connectedness in a general real closed field. In order to study the properties of closed and bounded semi-algebraic sets in Section 4, we introduce semi-algebraic germs in Section 3. The semi-algebraic germs over a real closed field constitute a real closed field containing infinitesimals, closely related to the field of Puiseux series, and play an important role throughout the whole book. We end the chapter with a section on semi-algebraic differentiable functions.
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