I just noticed this question about Lorentzian convergence for which some results have been obtained, albeit by far not as strong as in the metric case. In case anyone is interested in having a discussion about this, getting some references or some idea as to why it is somewhat more difficult than the standard metric theory, then he or she may contact me at my homepage http://be.linkedin.com/pub/johan-noldus/1b/416/889
Let me mention at this point already that to my best knowledge, I was the first person to develop a Lorentzian theory of convergence and that no equivalence with any metric theory of convergence exists (no use of time functions and so on). There is a paper by Sormani http://arxiv.org/abs/1006.0411 which mentions my first paper; I do not understand however why she mentions that no stability theorems are known in this context since I have explained my work to her http://www.ams.org/notices/200404/lawrenceville-prog.pdf on the occasion of an AMS meeting. Certainly, the paper on this http://arxiv.org/abs/gr-qc/0402049 was alreay on the web by then.
Obviously, a link may be provided to the answers I give at my homepage, but for general reasons I prefer to limit my writings to my homepage.
Kind regards,
Johan Noldus