Can a closed universe become open? My feeling is that no a closed universe can't become open. Under normal evolution according to the Friedman equation the curvature would be more exagerated as time goes on (i.e. a a closed universe becomes more closed). Under say inflation the curvature can be flattened but i hesitate to say that it can't be flipped (i.e. go from closed to open), is this correct? 
 A: Topologically speaking, open universes and closed universes have a different topology ($\mathbb{R}^3$ versus $S^3$). Topology change would require the introduction of something bad (either closed timelike curve, loss of time orientation or naked singularities), none of which are compatible with FRW spacetimes (in particular, homogeneous spacetimes must be geodesically complete). Unless you're willing to drop the homogeneity and isotropy of space, one cannot change into the other in a continuous manner.
Edit : but wait! Consider a FRW spacetime, initially open, with a radius diverging in finite time. This would imply that the spacetime is indeed not causally closed, which may still allow for topology change. On the other hand, I'm not sure this makes sense as a spacetime, as if we consider two observers at the two poles of $S^3$, they would become separated by an infinite distance in finite time (and this is true of all points by homogeneity). It's difficult to consider as I can't find any coordinates where this scenario would play out without problems.
