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?
Applications of non-Hausdorff spaces
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0What do you mean by applications. If you don't mind could please give some examples of applications of Hausdorff spaces! – 2012-07-08
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2The spectrum of a ring, given the Zariski topology, is almost never Hausdorff, and these spectra are of great importance in (modern) algebraic geometry. – 2012-07-08
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0I've been assured that finite topological spaces have many applications, and any finite topological space that is not discrete is not Hausdorff (it is not even $T_1$). – 2012-07-08
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0@Alex can you please tell me about some applications of finite non-hausdorff topological spaces? – 2012-07-08
2 Answers
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.
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!