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Specialization and a local homeomorphism theorem for real Riemann surfaces of rings. (English) Zbl 0868.13004
Let $$\varphi: k\to A$$ and $$f:A\to R$$ be ring morphisms, $$R$$ a real ring. We prove that if $$f: A\to R$$ is étale, then the corresponding mapping between real Riemann surfaces $S_r(f): S_r(R/k) \to S_r(A/k)$ is a local homeomorphism. Several preparatory results are proved, as well. The most relevant among these are:
(1) a Chevalley theorem for real Riemann surfaces on the preservation of constructibility via $$S_r(f)$$ and,
(2) an analysis of the closure operator on real Riemann surfaces.
Constructible sets are dealt with by means of a suitable language.

##### MSC:
 13A18 Valuations and their generalizations for commutative rings 13J25 Ordered rings 14A05 Relevant commutative algebra 12J10 Valued fields 12J15 Ordered fields 03C07 Basic properties of first-order languages and structures
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