TITLE: How should one think about non-Hausdorff topologies?
QUESTION [65 upvotes]: 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.

REPLY [5 votes]: Some rather strong non-Hausdorff topologies arise quite naturally in topological dynamics: orbit spaces of group actions, leaf spaces of foliations, etc. These spaces are not often studied in their own right, because the Hausdorff space of which they are the quotient is usually sitting right there in front of our eyes and is quite adequate for all purposes. 
However, in studying $Out(F_n)$, the outer automorphism group of a free group $F_n$ of rank $n$, a certain non-Hausdorff orbit space plays a central role in the theory that was first developed in the papers by Bestvina, Feighn, and Handel on the Tits alternative for $Out(F_n)$. The group $F_n$ may be identified with the fundamental group of some finite graph $G$, and with the deck transformation group of the universal cover $\tilde G$. Its Gromov boundary $\partial F_n$ is a Cantor set identified with the space of ends of $\tilde G$. The "space of lines upstairs", denoted $\tilde{\mathcal B}(F_n)$, is the space of all 2-point subsets $\partial F_n$ equipped with the Hausdorff topology, and may be identified with the "lines in $\tilde G$"---bi-infinite edge paths in $\tilde G$ without backtracking---equipped with the topology having a basis element for each finite path $\tilde\alpha$ in $\tilde G$ consisting of all lines containing $\tilde\alpha$. The left action of $F_n$ on itself, or its deck transformation action on $\tilde G$, extends continuously to an action on $\partial F_n$ and induces an action on $\tilde{\mathcal B}(F_n)$ whose non-Hausdorff orbit space $\mathcal B(F_n)$ is the "space of lines (downstairs)"; it may be identified with the set of bi-infinite edge paths in $G$ without backtracking (ignoring parameterization) equipped with the topology having a basis element for each finite path $\alpha$ in $G$ consisting of all lines having $\alpha$ as a subpath.
Here is some intuition used in understanding $\tilde{\mathcal B}(F_n)$. In analogy with Thurston's unstable geodesic laminations, an outer automorphism $\phi \in Out(F_n)$ can have an "attracting lamination", a subset $\Lambda \subset \tilde{\mathcal B}(F_n)$ which is the closure of a single point $\ell$ such that for some open set $U \subset \tilde{\mathcal B}(F_n)$, the forward iterates $\phi^i(U)$ form a neighborhood basis for $\ell$, plus additional properties on $\ell$ ruling out some simple pathological possibilities. This point $\ell$ is thought of as a "generic leaf" of $\Lambda$, and the properties of $\ell$ are often used as stand-ins for properties of $\Lambda$ as a whole. As a simple example: in the graph $G$, given an edge $E$, the line in $G$ realizing a given generic leaf $\ell$ crosses $E$ if and only if every generic leaf of $\Lambda$ crosses $E$, in which case one might wish say that $\Lambda$ itself crosses $E$. But it is important to draw the distinction between such and edge $E$ and some other edge $E'$ which is crossed by some nongeneric leaves of $\Lambda$ but not by any generic leaves. It sometimes happen that $\Lambda$ is "minimal" and so every leaf is generic and one's intuition is more easily developed; but the very real possibility of nongeneric leaves is important to understand as well. 
The non-Hausdorff space $\mathcal B(F_n)$ can indeed be described as a certain quotient of a certain Hausdorff space, but that Hausdorff space depends on rather unnatural choices: $\mathcal B(F_n)$ is a quotient of the space of parameterized bi-infinite edge paths in $G$ with the compact open topology. The unnaturality of this construction lies in the unnaturality of the choice of the graph $G$, which varies over the Culler-Vogtmann "outer space" of $F_n$. The graph $G$ is not well-defined up to homeomorphism, although it is well-defined up to homotopy equivalence. This unnaturality is perhaps one factor that led to the emergence of the non-Hausdorff space $\mathcal{B}(F_n)$ as the central object, rather than the space of bi-infinite bi-infinite edge paths in some choice of $G$. 
This situation is somewhat similar to what happens in the study of the mapping class group of a closed oriented surface $S$ of genus at least $2$. As one varies over all possible hyperbolic structures on $S$, the quotient space of the geodesic foliation of the projective line bundle of $S$ has an invariant definition independent of the choice of hyperbolic structure, namely it is the non-Hausdorff quotient space $\partial^2 \pi_1 S / \pi_1 S$. That "invariance" has never been particularly inviting to dynamicists, who would rather think about the geodesic foliation itself (or maybe the geodesic flow on the unit tangent bundle). But when Thurston developed his theory of geodesic laminations on $S$, he often took pains to point out that the space of geodesic laminations was well-defined independent of the somewhat unnatural choice of hyperbolic structure.