  We recast the tools of ``global causal analysis'' in accord with an approach
to the subject animated by two distinctive features: a thoroughgoing reliance
on order-theoretic concepts, and a utilization of the Vietoris topology for the
space of closed subsets of a compact set. We are led to work with a new causal
relation which we call $K^+$, and in terms of it we formulate extended
definitions of concepts like causal curve and global hyperbolicity. In
particular we prove that, in a spacetime $\M$ which is free of causal cycles,
one may define a causal curve simply as a compact connected subset of $\M$
which is linearly ordered by $K^+$. Our definitions all make sense for
arbitrary $C^0$ metrics (and even for certain metrics which fail to be
invertible in places). Using this feature, we prove for a general $C^0$ metric,
the familiar theorem that the space of causal curves between any two compact
subsets of a globally hyperbolic spacetime is compact. We feel that our
approach, in addition to yielding a more general theorem, simplifies and
clarifies the reasoning involved. Our results have application in a recent
positive energy theorem, and may also prove useful in the study of topology
change. We have tried to make our treatment self-contained by including proofs
of all the facts we use which are not widely available in reference works on
topology and differential geometry.
