I'm studing on "Real Algebraic Geometry" by Bochnak-Coste-Roy and in Chapter 11 I found a very interesting characterization of the Euler-Poincaré characteristic of real algebraic sets over a general real closed field. Now I'm a bit confused about the involved definition of homology, since in this book it is not defined. In particular I'm intereted in solving the following issues:
- In this setting the homology, with rational coefficients, is defined by mappings from affine simplices in $\mathbb{R^n}$, as usual, or in the affine space over the real closed field where the algebraic set is defined?
- In $\mathbb{R^n}$ I know that the simplicial homology of a finite simplicial complex and the singular homology of its realization do coincide, the proof can be found in "Elements of Algebraic Topology" by Munkres. In "Real Algebraic Geometry" they seem to apply this result also in a general real closed field without any comment, is there any reference where I can find the generalization of the previous result? I think it must work but, for instance, somewhere in the proof it may be used the compactness of simplices in $\mathbb{R^n}$ that is false in real closed fields in general. Thus, I am interested in understanding whether the proof of the previous result can be generalized or it needs another aproach.
I am really grateful to anyone would give an answer to my doubts.