  We characterize when the primitive ideal space of a crossed product $\acg$ of
a \cs-algebra $A$ by a locally compact abelian group $G$ is a $\sigma$-trivial
$\ghat G$-space for the dual $\ghat G$-action. Specifically, we show that
$\Prim(\acg)$ is $\sigma$-trivial if and only if the quasi-orbit space is
Hausdorff, the map which assigns to each quasi-orbit $\w$ a certain subgroup
$\ttg(\alpha^\w)$ of the Connes spectrum of the system $(A_\w,G,\alpha^\w)$ is
continuous, and there is a generalized Green twisting map for $(A,G,\alpha)$.
Our proof requires a substantial generalization of a theorem of Olesen and
Pedersen in which we show that there is a generalized Green twisting map for
$(A,G,\alpha)$ if and only if $\acg$ is isomorphic to a generalized induced
algebra.
