  We introduce the notion of combinatorial type of varieties $X$ which
generalizes the concept of the dual complex of SNC divisors. It is a unique, up
to homotopy, finite simplicial complex $\Sigma(X)$ which is functorial with
respect to morphisms of varieties. Its cohomology $H^i(\Sigma(X),Q)$ for
complex projective varieties coincide with weight zero part of the Deligne
filtration $W_0(H^i(X,Q))$. The notion can be understood as a topological
measure of the singularities of algebaric schemes of finite type. We also prove
that any variety in characteristic zero admits the Hironaka desingularization
with all fibers having SNC. Moreover the dual complexes of the fibers are
isomorphic on strata. Also for any morphism $f:X\to Y$ there exists a similar
desingularization $\tilde{X}\to X$ for which the induce morphism $\tilde{X}\to
Y$ has SNC fibers. One of the consequence is that for any projective morphism
$f:X\to Y$ the combinatorial type of the fiber is a constructible function. In
particular $\dim(W_0H^i(f^{-1}(y))$ is constructible.
