  Ultracold quantum gases are highly controllable and, thus, capable of
simulating difficult quantum many-body problems ranging from condensed matter
physics to astrophysics. Although experimental realizations have so far been
restricted to flat geometries, recently also curved quantum systems, with the
prospect of exploring tunable geometries, are produced in microgravity
facilities as ground-based experiments are technically limited. Here we analyze
bubble-trapped condensates, in which the atoms are confined on the surface of a
thin spherically-symmetric shell by means of external magnetic fields. A
thermally-induced proliferation of vorticity yields a vanishing of
superfluidity. We describe the occurrence of this topological transition by
conceptually extending the theory of Berezinskii, Kosterlitz and Thouless for
infinite uniform systems to such finite-size systems. Unexpectedly, we find
universal scaling relations for the mean critical temperature and the finite
width of the superfluid transition. Furthermore, we elucidate how they could be
experimentally observed in finite-temperature hydrodynamic excitations.
