Timeline for What is the meaning of non-Hausdorff spaces in algebraic geometry
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| Jan 4, 2015 at 23:05 | comment | added | Allen Knutson | Indeed, AG doesn't really care about topology. E.g. one is hardly interested in the set of all Zariski-continuous maps from one scheme to another, such as complex conjugation on $\mathbb C$. One extreme point of view on this (Grothendieck's, expressed in the letter where he introduced dessins d'enfants) is that general topology was a mistake from the get-go, as evidenced by theorems about $T_1$ spaces and whatnot, when what Poincar\'e wanted to study was manifolds. | |
| Jan 4, 2015 at 21:09 | comment | added | truebaran | Thank you for your answer: I agree that the coincidence of two properties: being closed and being subvariety is nice and I believe that it is convenient. But you said that you care more about closed sets rather than open: however topology is about "being close without measuring distances" so the central notion in topology is the notion of (small) neighbourhoods and the convergence. However in algebraic geometry open sets are always very big: so the question is whether algebraic geometry really cares about topology or maybe topology provides only a convenient way to formulate results in AG? | |
| Jan 4, 2015 at 19:37 | history | answered | Simon Rose | CC BY-SA 3.0 |