Non-Hausdorff Manifolds via Adjunction Spaces

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.