  Singularities play an important role in General Relativity and have been
shown to be an inherent feature of most physically reasonable space-times.
Despite this, there are many aspects of singularities that are not
qualitatively or quantitatively understood. The abstract boundary construction
of Scott and Szekeres has proven to be a flexible tool with which to study the
singular points of a manifold. The abstract boundary construction provides a
'boundary' for any n-dimensional, paracompact, connected, Hausdorff, smooth
manifold. Singularities may then be defined as entities in this boundary - the
abstract boundary. In this paper a topology is defined, for the first time, for
a manifold together with its abstract boundary. This topology, referred to as
the attached point topology, thereby provides us with a description of how the
abstract boundary is related to the underlying manifold. A number of
interesting properties of the topology are considered, and in particular, it is
demonstrated that the attached point topology is Hausdorff.
