  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.
