  Motivated by condensed matter physical systems, in which a finite-time
singularity indicates that the topology of the system changes, we critically
examine the possibility of the Universe's topology change at a finite-time
cosmological singularity. We emphasize on Big Rip and Type II and IV
cosmological singularities, which we classify to future spacelike and timelike
singularities. For the Type IV and Type II singularities, since no geodesics
incompleteness occurs, no topological change is allowed, by using Geroch's
theorem arguments. However, for the Big Rip case, Tipler's arguments allow a
topology change, if the spacetime in which the topology change occurs is
non-compact and the boundary of this region are two topologically distinct
three dimensional spacelike partial Cauchy hypersurfaces. Also, some additional
requirements must hold true, among which the weak energy condition, which can
be satisfied in a geometric way in the context of a modified gravity. We
critically examine Tipler's arguments for the Big Rip case, and we discuss the
mathematical implications of such a topological change, with regard to the
final hypersurface on which geodesics incompleteness occurs.
