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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?

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  • $\begingroup$ What do you mean by applications. If you don't mind could please give some examples of applications of Hausdorff spaces! $\endgroup$ – Mercy King Jul 8 '12 at 9:53
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    $\begingroup$ The spectrum of a ring, given the Zariski topology, is almost never Hausdorff, and these spectra are of great importance in (modern) algebraic geometry. $\endgroup$ – Ben Jul 8 '12 at 9:57
  • $\begingroup$ I'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 $T_1$). $\endgroup$ – Alex Becker Jul 8 '12 at 9:57
  • $\begingroup$ @Alex can you please tell me about some applications of finite non-hausdorff topological spaces? $\endgroup$ – K A Khan Jul 8 '12 at 10:15
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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.

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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!

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