Topological space associated to a real or complex scheme

Hi, consider a scheme $X$ of finite type over $\mathbb{R}$ (or $\mathbb{C}$). In Hartshorne's appendix B on 'transcendental methods' it is shortly mentioned how to assign a reasonable topological space to $X$ and he says that one can use the 'same glueing data'. My question is two-parted:

Does this construction actually define a functor $R_{\mathbb{R}}:Sch/\mathbb{R}\to Top$ from schemes of finite type over $\mathbb{R}$ to topological spaces? (I suppose that a positive/negative answer would hold for $\mathbb{C}$ also.)

And:

Does anyone know where this is written up rigorously, I mean without just saying that one just 'uses the same glueing data' and everything is fine?

Thank you!

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Yes (yes) and no I don't really know a good reference. – Donu Arapura Apr 12 '11 at 16:17
The space $R_{\mathbb{R}}(X)$ has points consisting of maps Spec $\mathbb{C} \rightarrow X$ over $\Spec \mathbb{R}$. So complex conjugation induces an involution. The natural target for $R_{\mathbb{R}}$ is $\mathbb{Z}/2$-spaces, i.e., spaces equipped with an action of $\mathbb{Z}/2$. – Dan Isaksen Apr 13 '11 at 11:04

As for the real algebraic geometry, the standard reference is Bochnak-Coste-Roy's Real Algebraic Geometry. What you want to study is Chapter 11 Topology of real algebraic varieties. Be warned that this is an advanced monograph and that general knowledge of schemes is insufficient : techniques like positivstellensatz, real spectra, Nash functions,...are de rigueur here. On the positive side you will be delighted by astonishing results like : every $\mathcal C^\infty$ compact manifold is diffeomorphic to an algebraic subvariety of some $\mathbb R ^p$ !