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?](https://mathoverflow.net/questions/26374/for-which-classes-of-topological-spaces-euler-characteristics-is-defined)



How to prove that a projective variety is a finite CW complex?
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).


