Rings of Continuous Functions as Real Closed Rings

Abstract

Real closed rings were first introduced in [48] (cf. [50], [51]) for the express purpose of establishing a new foundation for semi-algebraic geometry. The basic idea is to start with any semi-real ring A (cf. [29], p. 103, Definition 1), i.e., a ring with nonempty real spectrum Sper(A) ([29], Kapitel III, §3; for the real spectrum also see [4], Chapitre 7), and to associate with A its real closure ρ(A). The real closure should be thought of as the ring of continuous semi-algebraic functions on Sper(A). In a setting where the usual notion of continuity of functions is not applicable the real closure supplies in a categorical sense a largest ring of functions on Sper(A) which is useful for studying the topology of Sper(A). (Categorical properties of the real closure are exhibited in [36]. A development of the foundations of the algebraic topology of the real spectrum will be given in [57].) For the purposes of classical topology the largest partially ordered ring of functions on a space X to be used profitably is the ring C(X, ℝ) of continuous functions. Thus, the real closure is a substitute for the ring of continuous functions on any topological space. It is an obvious question whether this is a mere analogy, or whether there is a more formal connection between rings of continuous functions and real closures.