  In an earlier paper, we showed that the causal boundary of any homogeneous
plane wave satisfying the null convergence condition consists of a single null
curve. In Einstein-Hilbert gravity, this would include any homogeneous plane
wave satisfying the weak null energy condition. For conformally flat plane
waves such as the Penrose limit of $AdS_5 \times S^5$, all spacelike curves
that reach infinity also end on this boundary and the completion is Hausdorff.
However, the more generic case (including, e.g., the Penrose limits of $AdS_4
\times S^7$ and $AdS_7 \times S^4$) is more complicated. In one natural
topology, not all spacelike curves have limit points in the causal completion,
indicating the need to introduce additional points at `spacelike infinity'--the
endpoints of spacelike curves. We classify the distinct ways in which spacelike
curves can approach infinity, finding a {\it two}-dimensional set of distinct
limits. The dimensionality of the set of points at spacelike infinity is not,
however, fixed from this argument. In an alternative topology, the causal
completion is already compact, but the completion is non-Hausdorff.
