This is the updated version of the author’s Habilitationsschrift [“Real closed spaces” (1984; Zbl 0586.14016)] including a more general definition of affine real closed spaces.
Real closed spaces provide, using the theory of real spectra, an abstract generalization, in the case of general rings, of semi-algebraic sets equipped with semi-algebraic (continuous) functions.
Chapter 1 is devoted to the introduction of a closure operator on the category of schemes. Chapter 2 and 3 introduce and discuss real closed sheaves. Chapter 4 makes the connection with semi-algebraic geometry. In chapter 5 a general theory of real closed spaces is started. Using this theory more precise results on the connection between semi-algebraic geometry and affine real closed spaces are obtained in chapter 6.