Non-Hausdorff topology

In most topology books, the Hausdorff separation property is assumed from the very start and contain very little information on non-Hausdorff spaces. In classical mathematics, most topological spaces are indeed Hausdorff. But non-Hausdorff spaces are important already in algebraic geometry, and crucial in fields such as domain theory. Indeed, in connection with order, non-Hausdorff spaces, especially $T_0$ spaces, play a more significant role than Hausdorff spaces. Sobriety is probably the most important and useful property of non-Hausdorff topological spaces. It has been used in the characterizations of spectral spaces and $T_0$ spaces that are determined by their open set lattices. With the development of domain theory, another two properties also emerged as the very useful and important properties for non-Hausdorff topology theory: $d$-space and well-filtered space. In the past few years, some remarkable progresses have been achieved in understanding such structures. In this chapter, we shall make a brief survey on some of these progresses and list a few related problems.