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Motivated by condensed matter physical systems, in which a finite-timesingularity indicates that the topology of the system changes, we criticallyexamine the possibility of the Universe's topology change at a finite-timecosmological singularity. We emphasize on Big Rip and Type II and IVcosmological singularities, which we classify to future spacelike and timelikesingularities. For the Type IV and Type II singularities, since no geodesicsincompleteness occurs, no topological change is allowed, by using Geroch'stheorem arguments. However, for the Big Rip case, Tipler's arguments allow atopology change, if the spacetime in which the topology change occurs isnon-compact and the boundary of this region are two topologically distinctthree dimensional spacelike partial Cauchy hypersurfaces. Also, some additionalrequirements must hold true, among which the weak energy condition, which canbe satisfied in a geometric way in the context of a modified gravity. Wecritically examine Tipler's arguments for the Big Rip case, and we discuss themathematical implications of such a topological change, with regard to thefinal hypersurface on which geodesics incompleteness occurs.

Is a Topology Change After a Big Rip Possible?