How to prove that a projective variety is a finite CW complex? Let $X$ be a (singular) projective variety, in other words something given by a collection of polynomial equations in $\mathbb CP^n$ or $\mathbb RP^n$. How can one prove it is a finite $CW$ complex? 
Similar question: Suppose that $X$ affine (i.e. given by polynomial equations in $\mathbb C^n$, or $\mathbb R^n$). How can one prove its one point compactification is a finite $CW$ complex? 
These questions are sequel to the discussions here:
For which classes of topological spaces Euler characteristics is defined?
 A: The following thesis seems to be more general and worth referencing at this post:
Triangulation of Locally Semi-Algebraic Spaces. by K.R. Hofmann.
I quote from the abstract:

"We give necessary and sufficient conditions for a locally
  semi-algebraic space to be homeomorphic to a simplicial complex. Our
  proof does not require the space to be embedded anywhere, and it
  requires neither compactness nor projectivity of the space. A
  corollary is that every real or complex algebraic variety is
  triangulable, a result which does not seem to be available in the
  literature when the variety is neither projective nor real and
  compact."

A: The Lojasiewicz theorem says that every semi-algebraic subset of $\mathbf{R}^n$ can be triangulated. Moreover, there is a similar statement for pairs of the form (a semi-algebraic set, a closed subset). See e.g. Hironaka, Triangulations of algebraic sets, Arcata proceedings 1974 and references therein (including the original paper by Lojasiewicz).
The case of an arbitrary (not necessarily quasi-projective) complex algebraic variety follows from Nagata's theorem (every variety can be completed) and Chow's lemma (every complete variety can be blown up to a projective one).
