  The {\em abstract boundary\/} (or {\em {\em a\/}-boundary\/}) of Scott and
Szekeres \cite{Scott94} constitutes a ``boundary'' to any $n$-dimensional,
paracompact, connected, Hausdorff, $C^\infty$-manifold (without a boundary in
the usual sense). In general relativity one deals with a {\em space-% time\/}
$(\cM,g)$ (a 4-dimensional manifold $\cM$ with a Lorentzian metric $g$),
together with a chosen preferred class of curves in $\cM$. In this case the
{\em a\/}-boundary points may represent ``singularities'' or ``points at
infinity''. Since the {\em a\/}-boundary itself, however, does not depend on
the existence of further structure on the manifold such as a Lorentzian metric
or connection, it is possible for it to be used in many contexts.
  In this paper we develop some purely topological properties of abstract
boundary sets and abstract boundary points ({\em a\/}-boundary points). We
prove, amongst other things, that compactness is invariant under boundary set
equivalence, and introduce another invariant concept ({\em isolation\/}), which
encapsulates the notion that a boundary set is ``separated'' from other
boundary points of the same embedding. .......
  [The abstract continues in paper proper - truncated to fit here.]
