Applications of non-Hausdorff spaces

Question

Mostly the topological spaces are considered Hausdorff while working in topology and geometry. I came to know about the uses of non-Hausdorff spaces in algebraic geometry (However I don't know How!). Can someone give me some idea of applications of non-Hausdorff spaces?

Answer 1

A very natural class of examples are the orbit spaces of any non-free continuous group actions. These spaces are good to study, because they show the underlying structure of the action.

For example, look at the height function on $S^1$, and the gradient flow $\phi$. Put an equivalence relation on $S^1$, $x\cong y$ if there exists $t\in \mathbb{R}$ such that $\phi_t(x)=y$ (that is, you can reach one point by the other following the flow). The space $S^1/\cong$ consists of $4$ points, two points corresponding to the equilibria, and two points corresponding to the connecting orbits. The topology on this space is non-Hausdorff. It is a good idea to work out all the open sets in this example by hand. If you need help with this, please don't hesitate to ask.

Answer 2

An example of something useful that very rarely is Hausdorff is the espace étalé in sheaf theory. Sheaf theory has applications to algebraic geometry, algebraic topology and even engineering!