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, Ank, 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 Ank 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?

**Answer**

The digital line is a non-Hausdorff space important in graphics. The underlying set of points is just 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 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 Z which is T0 but not T1 (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.

**Attribution***Source : Link , Question Author : Eric , Answer Author : Randall*