Abstract

In this talk we will introduce various aspects of the weird world of non-Hausdorff manifolds. Typically the Hausdorff property is included in the definition of a manifold for technical convenience. The alternative is somewhat daunting: without the Hausdorff property we do not have access to partitions of unity in their full generality, and thus various structures may or may not exist in the non-Hausdorff case. However, as we will see, certain topological representations allow us to circumvent this issue and recreate differential geometry without the need of arbitrarily-existent partitions of unity. To illustrate this idea, we will start from the topology of non-Hausdorff manifolds and then slowly introduce more and more structure of various interest. I will try to keep the talk self-contained, though I will need to assume familiarity with the basics of differential geometry and point-set topology.