Let $A$ be a $C^*$-algebras with ideal space $\mathcal{I} (A)$ and equip $\mathcal{I} (A)$ with the Fell topology, i.e. the topology generated by the subbase $U_{I}:=\left\{ J\in{\cal I}\left(A\right)\mid I\nsubseteq J\right\}$, $I \in \mathcal{I} (A)$. In this paper, on page 2, the author states that if $B$ is another $C^*$-algebra and $\varphi: \mathcal{I}(A)\rightarrow \mathcal{I}(B)$ is a function which preserves the inclusion of ideals, then $\varphi$ must be continuous.
Maybe this statement is obvious but I failed to show it rigorously so far. Can somebody help?
I think what they are saying is simply that $$ \varphi^{-1}(U_I)=\{H:\ \varphi(H)\in U_I\} =\{H:\ I\subsetneq\varphi(H)\} =\{H:\ \varphi^{-1}(I)\subsetneq H\} =U_{\varphi^{-1}(I)}. $$