TITLE: "Naturally occurring" non-Hausdorff spaces?
QUESTION [75 upvotes]: It is not difficult for a beginning point-set topology student to cook up an example of a non-Hausdorff space; perhaps the simplest example is the line with two origins. It is impossible to separate the two origins with disjoint open sets.
It is also easy for a beginning algebraic geometry student to give a less artificial example of a non-Hausdorff space: the Zariski topology on affine $n$-space over an infinite field $k$, $\mathbf{A}_{k}^{n}$, is not Hausdorff, due to the fact that polynomials are determined by their local behavior. Open sets here are in fact dense.
I am interested in examples of the latter form. The Zariski topology on $\mathbf{A}_{k}^{n}$ exists as a tool in its own right, and happens to be non-Hausdorff. As far as I'm aware, the line with two origins doesn't serve this purpose. What are some non-Hausdorff topological spaces that aren't merely pathological curiosities?

REPLY [8 votes]: There are finite spaces that model the homotopy theory of finite simplicial complexes.  Since simplicial complexes can model so much of what a topologist might think of as the "good" spaces for doing homotopy theory, it might be a bit of a surprise that finite spaces suffice in some sense.
It essentially began with McCord and Stong in 1966.  (Interestingly, they came about these results nearly independently and at the same time:  Stong mentions McCord's result as a preprint in his paper.)  At any rate, McCord proved the following, the proof of which is constructive:
Theorem:  Given any finite simplicial complex $K$, there exists a finite topological space $X$ and a weak homotopy equivalence $f: |K| \to X$.  The converse holds as well (starting with $X$, constructing $K$).
Of course, $X$ will be non-Hausdorff as soon as it is not discrete (as all the interesting ones are).
Hence, finite spaces have the same homotopy and singular homology groups as finite simplicial complexes.  In particular, there are finite spaces weakly homotopy equivalent to the $n$-spheres.  This finiteness might give one some more control over the behavior of the spaces involved.  This idea has been pushed by Hardie, Witbooi, et al, who have been able to find finite models for decidely-not-finite constructions like the Hopf map $\eta: S^3 \to S^2$ and the non-trivial element of $\pi_5(S^3)$.  There may be several more recent applications that I am not aware of since getting old.

McCord, Michael C. Singular homology groups and homotopy groups of finite topological spaces. Duke Math. J. 33 (1966), 465–474.
Stong, R. E. Finite topological spaces. Trans. Amer. Math. Soc. 123 (1966), 325–340.
Hardie, K. A.; Vermeulen, J. J. C.; Witbooi, P. J. A nontrivial pairing of finite $T_0$ spaces. Topology Appl. 125 (2002), no. 3, 533–542.
Hardie, K. A.; Witbooi, P. J. Crown multiplications and a higher order Hopf construction. Topology Appl. 154 (2007), no. 10, 2073–2080.