The purpose of this paper is to present results and open problems related to R-places. The first section recalls basic facts, the second introduces R-places and their relationship with orderings and valuations. The third part involves Real Algebraic Geometry and gives results proved using the space of R-places. Theorem 14 gives explicitly, in terms of the function field of the variety, the number of connected components of a non-empty smooth projective real variety. The fourth and fifth parts are devoted to the links with the real holomorphy rings and the valuation fans. Then we present an approach to abstract real places and conclude with some open questions.

中文翻译：

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Nombre de composantes connexes d'une variete reelle et R−places

本文的目的是展示与 R-places 相关的结果和未解决的问题。第一部分回顾基本事实，第二部分介绍 R 位及其与排序和估值的关系。第三部分涉及实代数几何，并给出了使用R位空间证明的结果。定理 14 在簇的函数域方面明确给出了非空光滑射影实簇的连通分量数。第四和第五部分专门讨论与真实全息环和估值爱好者的联系。然后我们提出了一种抽象真实地点的方法，并以一些开放性问题作为结论。