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?
Best Answer
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.