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!