  A pure topological characterization of primitive ideal spaces of separable
nuclear C*-algebras is given. We show that a $T_0$-space $X$ is a primitive
ideal space of a separable nuclear C*-algebra $A$ if and only if $X$ is
point-complete second countable, and there is a continuous pseudo-open and
pseudo-epimorphic map from a locally compact Polish space $P$ into $X$.
  We use this pseudo-open map to construct a Hilbert bi-module $\mathcal{H}$
over $C_0(X)$ such that $X$ is isomorphic to the primitive ideal space of the
Cuntz--Pimsner algebra $\mathcal{O}_\mathcal{H}$ generated by $\mathcal{H}$.
Moreover, our $\mathcal{O}_\mathcal{H}$ is $KK(X;.,.)$-equivalent to $C_0(P)$
(with the action of $X$ on $C_0(P)$ given be the natural map from
$\mathbb{O}(X)$ into $\mathbb{O}(P)$, which is isomorphic to the ideal lattice
of $C_0(P)$.
  Our construction becomes almost functorial in $X$ if we tensor
$\mathcal{O}_\mathcal{H}$ with the Cuntz algebra $\mathcal{O}_2$.
