Non-Hausdorff Manifolds via Adjunction Spaces

TL;DR

This paper introduces a formalism for constructing non-Hausdorff manifolds by gluing Hausdorff manifolds along open submanifolds, analyzing where Hausdorff properties fail, and characterizing maximal Hausdorff submanifolds within them.

Contribution

It develops a theory of adjunction spaces for non-Hausdorff manifolds, providing a method to construct and analyze such spaces from Hausdorff manifolds.

Findings

Non-Hausdorff manifolds can be constructed by gluing along boundaries.

Hausdorff violation occurs precisely at the glued boundaries.

Partial characterization of maximal Hausdorff submanifolds within non-Hausdorff manifolds.

Abstract

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.