"Naturally occurring" non-Hausdorff spaces?

The digital line is a non-Hausdorff space important in graphics. The underlying set of points is just $\mathbb{Z}$. We give this the digital topology by specifying a basis for the topology. If $n$ is odd, we let $\{n\}$ be a basic open set. If $n$ is even, we let $\{n-1,n,n+1\}$ be basic open. These basic open sets give a topology on $\mathbb{Z}$, the resulting space being the "digital line." The idea is the odd integers $n$ give $\{n\}$ the status of a pixel, whereas the even $n$ encode $\{n-1,n,n+1\}$ as pixel-boundary-pixel. Thus this is a sort of pixelated version of the real line.

At any rate, this gives a topology on $\mathbb{Z}$ which is $T_0$ but not $T_1$ (and hence non-Hausdorff). That it is not Hausdorff is clear, since there is no way to separate $2$ from $3$. It also has tons of other interesting properties, such as being path connected, Alexandrov, and has homotopy and isometry similarities to the ordinary real line.

References added:

R. Kopperman T.Y. Kong and P.R. Meyer, A topological approach to digital topology, American Mathematical Monthly 98 (1991), no. 10, 901-917.

Special issue on digital topology. Edited by T. Y. Kong, R. Kopperman and P. R. Meyer. Topology Appl. 46 (1992), no. 3. Elsevier Science B.V., Amsterdam, 1992. pp. i–ii and 173–303.

Colin Adams and Robert Franzosa, Introduction to topology: Pure and applied, Pearson Prentice Hall, 2008.

In computer science, more specifically in programming languages theory, it is common to describe the behavior of computer programs using Scott-continuous functions, i.e. continuous functions between two topological spaces having the Scott topology. Roughly put, these are partially ordered sets where open sets are upper sets (if a point is in the set, all the larger points must be as well) which are inaccessible by directed suprema (the supremum of points outside the open set must also be outside). Equivalently: closed sets are lower sets closed under directed suprema.

This topology is in general not Hausdorff, since if $x < y$ any open set including $x$ must also contain $y$. More precisely, it is Hausdorff iff the order is trivial.

The Scott topology is a very important tool to provide a rigorous meaning to programs that may fail to terminate, either because they can get stuck in an infinite loop or in an infinite chain of recursive calls. The meaning of such program constructs is typically defined exploiting the Kleene fixed point theorem so to solve the "recursive equations" which arise from the self-referent (recursive) program.