TITLE: What application is there for a non-Hausdorff topological space?
QUESTION [9 upvotes]: I'm learning basic topology and as I understand it, a good way to intuit what an open set is, is that it determines which elements are near each other.  However, in a non-Hausdorff space, it would be possible for one point to be "near" another, without the reverse being true. For example, if $X=\{a,b\}$ and the typology $T=\{\emptyset, \{a\},X\}$ then "getting close to $b$" implies getting close to $a$, but not the reverse. You can get so close to $a$ that you are no longer close to $b$.
I know that these imprecise English words like "close" and "neighborhood" shouldn't be regarded as too strongly related to topological definitions, but I'm still wondering if it's possible to give an intuitive conception of what open sets "mean" or what they are, in a non-Hausdorff space.
[Edit: Giving examples of applications, particularly ones accessible to a person who has an beginner's acquaintance with topology, would be appreciated since they can illustrate the meaning of the concept.]

REPLY [2 votes]: Consider the closed unit sphere in 3-dim space and the function $f$ from the sphere to the 2-point space $\{a,b\}$ sending the interior of the sphere to $a$ and the boundary of the sphere into $b$. Under which topology on the target space is this function continuous? Well, since $f^{-1}(a)$ is an open subset of the sphere then we want $\{a\}$ to be open and this is the only requirement. The non Hausdorff topology allows you to treat this identification map as a continuous function between two topological spaces, allowing for the whole machinery to apply. Or, in other terms, allows you to think at the sphere, for all purposes in which inner points may be identified one with the other and boundary points may be identified one with the other, as a 2-point space with non Hausdorff topology.
A typical application of this arises each time you have a topological space $X$ and an equivalence relation ${\cal R}$ on it. Then you have a canonical way of endowing the set of equivalence classes with a topology. So you can say when two equivalence classes are "close to each other"... Take the most typical example in undergraduate geometry: classification of real conics under affine transformations. Since affine classes of conics are classified by rank and signature of the corresponding symmetric matrix you end up with a finite non Hausdorff topological space. Its topology tells you exactly which kind of transition from one conic to another may happen when you have a continous family of conics, so for example governs transitions of conic type inside linear families of conics.
As was said above group actions falls into this situation, and non Hausdorff spaces typically arises as spaces of orbits under group action.