  A method is presented for imputing a topology for any chronological set,
i.e., a set with a chronology relation, such as a spacetime or a spacetime with
some sort of boundary. This topology is shown to have several good properties,
such as replicating the manifold topology for a spacetime and replicating the
expected topology for some simple examples of spacetime-with-boundary; it also
allows for a complete categorical characterization, in topological categories,
of the Future Causal Boundary construction of Geroch, Kronheimer, and Penrose,
showing that construction to have a universal property for future-completing
chronological sets with spacelike boundaries. Rigidity results are given for
any reasonable future completion of a spacetime, in terms of the GKP boundary:
In the imputed topology, any such boundary must be homeomorphic to the GKP
boundary (if all points have indecomposable pasts) or to a topological quotient
of a closely related boundary (if boundaries are spacelike). A large class of
warped-product-type spacetimes with spacelike boundaries is examined,
calculating the GKP and other possible boundaries, and showing that the imputed
topology gives expected results; included among these are the Schwarzschild
singularity and those Robertson-Walker singularities which are spacelike.
