  We show that the zip stratification given by an arbitrary $\Ghat$-zip over a
scheme is pure. We deduce purity of the level-$m$-stratification for truncated
Barsotti-Tate groups and purity of the Ekedahl-Oort stratification for special
fibers of good models of Shimura varieties of Hodge type. We also show that all
Ekedahl-Oort strata are quasi-affine schemes and generalize several properties
of the Bruhat stratification from the PEL case to the case of Shimura varieties
of Hodge type.
