It's been known since Oppenheimer and Snyder's work in 1939 that it's easy to get a naked (i.e., timelike) singularity in models of spherically symmetric gravitational collapse, for forms of matter such as dust that obey all the standard energy conditions. (A review article on this kind of thing is Joshi 2011.) Whether this is physically realistic, stable with respect to perturbations, and so on is a different question, but not relevant for the purposes of this question.
Now it seems to me that the formation of a naked singularity by gravitational collapse is an example of topology change. Spacelike slices before the collapse have the trivial topology, while slices after the collapse have a hole in them at the singularity. (This is not the case for a black hole singularity, since a black hole singularity is spacelike.)
Topology change in GR has also been studied for a long time, and the classic reference seems to be Geroch 1967, which is summarized in Borde 1994. Geroch proves that topology change always involves both acausality and violation of the weak energy condition (WEC).
This confuses me, because doesn't dust satisfy the WEC? I'm sure I'm misunderstanding something, but I don't know what it is.
Borde, 1994, "Topology Change in Classical General Relativity," http://arxiv.org/abs/gr-qc/9406053
Geroch 1967, http://adsabs.harvard.edu/abs/1967JMP.....8..782G , paywalled
Joshi and Malafarina, "Recent developments in gravitational collapse and spacetime singularities," 2011, https://arxiv.org/abs/1201.3660