  In this paper we will introduce and develop a theory of adjunction spaces
which allows the construction of non-Hausdorff topological manifolds from
standard Hausdorff ones. This is done by gluing Hausdorff manifolds along
homeomorphic open submanifolds whilst leaving the boundaries of these regions
unidentified. In the case that these gluing regions have homeomorphic
boundaries, it is shown that Hausdorff violation occurs precisely at these
boundaries. We then use this adjunction formalism to provide a partial
characterisation of the maximal Hausdorff submanifolds that a given
non-Hausdorff manifold may admit.
