  We show that the weight 0 part of the first cohomology of a complex algebraic
variety $X$ is a topological invariant, and give an explicit description of its
dimension using a topological construction of the normalization of $X$, where
$X$ can be reducible, but must be equidimensional. The first assertion is known
in the $X$ compact case by A. Weber, where intersection cohomology is used.
Note that the weight 1 or 2 part of the first cohomology is not a topological
(or even analytic) invariant in the non-compact case by Serre's example.
