I have some problem when reading Theorem 3.6 of Kollar's book Shafarevich Maps and Automorphic Forms, page 41 (Corollary 3.5 of this article ), which states that
Let $X$ be a normal variety, $H\triangleleft \pi_1(X)$ a normal subgroup. Then
(i) there are open subsets $X^0\subset X$ and $Z^0\subset Sh^H(X)$ such that the H-shafarevich map $sh_X^H:X^0\to Z^0$ is everywhere defined,
(ii) every fiber of $sh_X^H|X^0$ is closed in $X$, and
(iii) $sh_X^H|X^0$ is a toplogically locally trivial fibration.
Here is a sketch: Pick $x\in VG(X)$, $w_0:W_0\to X$ normal cycle of maximal dimension, and $im[\pi_1(W_0)\to \pi_1(X)]\lesssim H$. we can find a locally topologically trivial family of normal cycles $S\longleftarrow W\longrightarrow X$ which contains $W_0$ such that $\text{im}\,w_s\ne \text{im}\, w_{s'}$. Our aim is to prove that the existence of $S^0\subset S$ such that $w: p^{-1}(S^0)\to X$ is an open immersion.
because a very general point of $X$ has only one pre image in $W$, therefore $w$ is birational. Let the $D_w$ be closed subset which is not immersion. Then it says that we can find $S_0$ and $D_s$ which is disjoint. Finally the proof ends with some settings.
My problem is: Why can't $D_w$ cover all of W? Why can we find the open set? Why can't the bad things cover all of $S$?