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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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    What 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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    The 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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    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 $T_1$).2012-07-08
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    @Alex can you please tell me about some applications of finite non-hausdorff topological spaces?2012-07-08

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