  In this paper we describe the topology of 4-dimensional closed orientable
Riemannian manifolds with a uniform lower bound of sectional curvature and with
a uniform upper bound of diameter which collapse to metric spaces of lower
dimensions. This enables us to understand the set of homeomorphism classes of
closed orientable 4-manifolds with those geometric bounds on curvature and
diameter. In the course of the proof of the above results, we obtain the soul
theorem for 4-dimensional complete noncompact Alexandrov spaces with
nonnegative curvature. A metric classification for 3-dimensional complete
Alexandrov spaces with nonnegative curvature is also given.
