  We show that Bunce-Deddens algebras, which are AT-algebras, are also limits
of circle algebras for Rieffel's quantum Gromov-Hausdorff distance, and
moreover, form a continuous family indexed by the Baire space. To this end, we
endow Bunce-Deddens algebras with a quantum metric structure, a step which
requires that we reconcile the constructions of the Latremoliere's
Gromov-Hausdorff propinquity and Rieffel's quantum Gromov-Hausdorff distance
when working on order-unit quantum metric spaces. This work thus continues the
study of the connection between inductive limits and metric limits.
