  When studying the causal propagation of a field in a globally hyperbolic
spacetime M, one often wants to express the physical intuition that it has
compact support in spacelike directions, or that its support is a spacelike
compact set. We compare a number of logically distinct formulations of this
idea, and of the complementary idea of timelike compactness, and we clarify
their interrelations. E.g., a closed subset A of M has a compact intersection
with all Cauchy surfaces if and only if A is contained in J(K) for some compact
set K. (However, it does not suffice to consider only those Cauchy surfaces
that partake in a given foliation of M.) Similarly, a closed subset A of M is
contained in a region between two Cauchy surfaces if and only if the
intersection of A with J(K) is compact for all compact K. We also treat future
and past compact sets in a similar way.
