The ideal space $\mbox{Id}(A)$ of a Banach algebra $A$ is studied as a bitopological space $(\mbox{Id}(A),\tau_u,\tau_n)$, where $\tau_u$ is the weakest topology for which all the norm functions $I\to\Vert a+I\Vert$ (with $a\in A$ and $I\in \mbox{Id}(A)$) are upper semi-continuous, and $\tau_n$ is the de Groot dual of $\tau_u$. When $A$ is separable, $\tau_n\vee\tau_u$ is either a compact, metrizable topology, or it is neither Hausdorff nor first countable. TAF-algebras are shown to exhibit the first type of behaviour. Applications to Banach bundles (which motivate the study), and to PI-Banach algebras, are given.