Fell topologies for AF-algebras and the quantum propinquity

We introduce a topology on the ideal space of any C∗-inductive limit built by an inverse limit of topologies and produce conditions for when this topology agrees with the Fell topology. With this topology, we impart criteria for when convergence of ideals of an AF-algebra can provide convergence of quotients in the quantum Gromov--Hausdorff propinquity building from previous joint work with Latr\'{e}moli\`{e}re. This bestows a continuous map from a class of ideals of the Boca--Mundici AF-algebra equipped with various topologies, including Jacobson and Fell topologies, to the space of quotients equipped with the propinquity topology.