Hausdorff closed limits and rigidity in Lorentzian geometry

We begin with a basic exploration of the (point-set topological) notion of Hausdorff closed limits in the spacetime setting. Specifically, we show that this notion of limit is well suited to sequences of achronal sets, and use this to generalize the `achronal limits' introduced in [12]. This, in turn, allows for a broad generalization of the notion of Lorentzian horosphere introduced in [12]. We prove a new rigidity result for such horospheres, which in a sense encodes various spacetime splitting results, including the basic Lorentzian splitting theorem. We use this to give a partial proof of the Bartnik splitting conjecture, under a new condition involving past and future Cauchy horospheres, which is weaker than those considered in [10] and [12]. We close with some observations on spacetimes with spacelike causal boundary, including a rigidity result in the positive cosmological constant case.