Algebraic Varieties vs Smooth Manifolds There are many posts I have read on that subject which seem unclear for me. My main question (it might be silly) is: 

"Every non-singular algebraic variety over $\mathbb{C}$ is a smooth
  manifold."

(see: http://mathoverflow.net/questions/7439/algebraic-varieties-which-are-also-manifolds) 
How? For an algebraic variety we have the Zariski topology which is not even Hausdorff? How can they be diffeomorphic then?

The statements it that if the variety is endowed with the relative topology as a subset of $\Bbb C^n$ then it is a manifold.
