Let $X$ be a scheme over $\mathbb{C}$. 


1. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
2. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the *weak* homotopy type of a finite CW-complex, (i.e. when is it a finite space)?


By "when", I mean what adjectives do I have to add to make this true, e.g. finite type, separated, smooth...


3. I'm also interested in question 1.) for when $X$ is affine.


If you happen to know a reference also, that would be fantastic. Thanks!


P.S. I'm aware of this mathoverflow question:


[How to prove that a projective variety is a finite CW complex?](https://mathoverflow.net/questions/26927/how-to-prove-that-a-projective-variety-is-a-finite-cw-complex)


However, it addresses only the case of varieties, unless I am missing something..



Homotopy types of schemes
Any scheme which is separated of finite type, has at least a triangulation, hence is, in particular, a CW-complex. In fact, by a theorem of [Lojasiewicz](http://www.numdam.org/numdam-bin/fitem?id=ASNSP_1964_3_18_4_449_0), this is true for any semi-algebraic set (one can even get this for subanalytic sets, by a result of Hironaka, in *Triangulation 
of algebraic sets*, Proc. Amer. Math. Soc. Inst. Algebra Geom. Arcata(1974)); however, the case of (possibly singular) algebraic varieties goes back to the early times of Algebraic Topology: e.g. these papers of [van der Waerden](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0102&DMDID=DMDLOG_0019) and of [Lefschetz and Whitehead](http://www.ams.org/journals/tran/1933-035-02/S0002-9947-1933-1501698-X/)). 


If you only are interested in weak homotopy types, it follows from Lurie's proper base change theorem that considering complex points satisfies proper (hyper)descent (this is Prop. 3.21 in this [paper of A. Blanc](http://arxiv.org/abs/1211.7360), which is now published in Compositio Math.). Using Hironaka's resolution of singularities theorem, this implies that, for any scheme of finite type $X$, the space $X(\mathbf{C})$ is a finite homotopy colimit of spaces of the form $Y(\mathbf{C})$ with $Y$ affine and smooth (using Mayer-Vietoris-like homotopy pushouts associated to blow-ups and to coverings by Zariski open subschemes). A smooth affine algebraic variety has the homotopy type of a finite CW-complex: this follows from Morse theory, as can be seen from (the proof of) Theorem 7.2, page 39 in Milnor's book *Morse Theory*.


From all this, we get that a sufficient condition for $X(\mathbf{C})$ to have the weak homotopy type of a finite CW-complex is to be of finite type, while a sufficient condition to get the homotopy type of a finite CW-complex is to be separated of finite type. A sufficient condition to get an actual finite triangulated space is to be proper.


If we drop the assumption that the scheme is of finite type, I don't see how we can control/define what happens unless we work with pro-homotopy types of some sort.


