  In this paper, we completely settle several of the open questions regarding
the relationships between the three most fundamental forms of set convergence.
In particular, it is shown that Wijsman and slice convergence coincide
precisely when the weak star and norm topologies agree on the dual sphere.
Consequently, a weakly compactly generated Banach space admits a dense set of
norms for which Wijsman and slice convergence coincide if and only if it is an
Asplund space. We also show that Wijsman convergence implies Mosco convergence
precisely when the weak star and Mackey topologies coincide on the dual sphere.
A corollary of these results is that given a fixed norm on an Asplund space,
Wijsman and slice convergence coincide if and only if Wijsman convergence
implies Mosco convergence.
