In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" (sequences/nets have unique limits, compact sets are closed, etc.). Most topological spaces encountered in undergraduate studies are indeed Hausdorff, often even normed or metrizable. However, at some point one finds that non-Hausdorff spaces do come up in practice, e.g. the Zariski topology in algebraic geometry, the Fell topology in representation theory, the hull-kernel topology in the theory of C\*-algebras, etc.


My question is: how should one think about (and work with) these topologies? I find it very difficult to think of such topological spaces as geometric objects, due to the lack of the intuitive Hausdorff axiom (and its natural consequences). With Hausdorff spaces, I often have some clear, geometric picture in my head of what I'm trying to prove and this picture gives good intuition to the problem at hand. With non-Hausdorff spaces, this geometric picture is not always helpful and in fact relying on it may lead to false results. This makes it difficult (for me, at least) to work with such topologies.


As this question is somewhat ambiguous, I guess I should make it a community wiki.


**EDIT**: Thanks for the replies! I got many good answers. It is unfortunate that I can accept just one.



How should one think about non-Hausdorff topologies?
For a variety of reasons, it's often useful to develop an intuition for *finite* topological spaces. Since the only Hausdorff finite spaces are discrete, one will have to deal with the non-Hausdorff case almost all the time. 


The fact of the matter is that the category of finite spaces is equivalent to the category of finite preorders, i.e., finite sets equipped with a reflexive transitive relation. In terms of a picture, draw an arrow $x \to y$ between points $x$ and $y$ whenever $x$ belongs to the closure of $y$ (or the closure of $x$ is contained in the closure of $y$). This defines a reflexive transitive relation. 


Two points $x$, $y$ have the same open neighborhoods if and only if $x \to y$ and $y \to x$. It follows that the topology is $T\_0$ (the topology can distinguish points) if and only if the preorder is a poset, where antisymmetry of $\to$ is satisfied. 


The closure of a point $y$ is the down-set {$x: x \to y$}, and a set is closed iff it is downward closed in the preorder. In the finite case, I believe it is true that every closed *irreducible* set (one that isn't the union of two proper closed subsets) is the closure of a point = principal ideal; if the point is unique, the space is called *sober*. Sober spaces are the kinds of spaces that arise as underlying topological spaces of schemes, and it seems to be true that a finite space is sober iff it is $T\_0$. 


