# “Naturally occurring” non-Hausdorff spaces?

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 $$nn$$-space over an infinite field $$kk$$, $$Ank\mathbf{A}_{k}^{n}$$, 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 $$Ank\mathbf{A}_{k}^{n}$$ 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?

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

At any rate, this gives a topology on $$Z\mathbb{Z}$$ which is $$T0T_0$$ but not $$T1T_1$$ (and hence non-Hausdorff). That it is not Hausdorff is clear, since there is no way to separate $$22$$ from $$33$$. 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.