Homotopy types of schemes Let $X$ be a scheme over $\mathbb{C}$. 


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*When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?

*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...


*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?
However, it addresses only the case of varieties, unless I am missing something..
 A: Let $X$ be a scheme over $\mathbf{C}$. I think we should endow $X(\mathbf{C})$ with a topology as follows. If $X = \text{Spec}(A)$ is affine, then we write $A = \text{colim}\ A_i$ with $A_i$ of finite type over $\mathbf{C}$, so $X = \lim X_i$ with $X_i = \text{Spec}(A_i)$. Then $X(\mathbf{C}) = \lim X_i(\mathbf{C})$ and we endow the left hand side with the limit topology where each $X_i(\mathbf{C})$ is endowed with the usual one. This just means that a set is open if it comes from an open in one of the $X_i(\mathbf{C})$. In general we glue these topologies; I think this obviously works but I didn't check the details.
Some weird things can happen here, for example it can happen that $X(\mathbf{C})$ is empty even though $X$ is not empty. Also, you can get $X(\mathbf{C})$ to be homeomorphic to any profinite space you like for affine $X$. If $A = S^{-1}\mathbf{C}[x, y]$ then you get $\mathbf{C}^2$ where you remove (possibly infinitely many) plane curves; so you can get $U = \{(x, y) \in \mathbf{C}^2 \mid x \not \in \mathbf{Z}\}$ for example.
All I am trying to say here is that we should probably require $X$ to be (at least) locally of finite type over $\mathbf{C}$. Not an answer.
A: 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, 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 and of Lefschetz and Whitehead). 
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, 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.
