Timeline for Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)
Current License: CC BY-SA 2.5
8 events
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| Jul 6, 2010 at 19:23 | comment | added | Kevin H. Lin | @Andrew: Back when you originally posted this answer, I did not really like it very much, but now rereading it several months later, I like it a lot! Especially the statement: "The language is fundamentally topological so there's no surprise at all that topological spaces result". | |
| Feb 10, 2010 at 9:35 | comment | added | Andrew Stacey | That's the point of bridges: they go both ways. Indeed, one could say that the whole of "algebraic topology" is the bridge in the other direction. | |
| Feb 10, 2010 at 2:01 | comment | added | Harry Gindi | Yeah, but functors of points pull a lot of the geometry back out of geometry. =p. | |
| Feb 9, 2010 at 21:31 | comment | added | Andrew Stacey | I might admit to going just a little over the top at the end ... | |
| Feb 9, 2010 at 21:15 | comment | added | Ilya Grigoriev | I agree with everything except for the "interesting = topological". I think it's more like "geometric implies topological". I'm a geometric person myself, but I think there are also perfectly fine - and interesting - non-geometric ways to look at problems; elementary Galois theory comes to mind (I don't just mean the Fundamental Theorem, but the whole philosophy of factoring polynomials by thinking of field extensions). | |
| Feb 9, 2010 at 16:09 | comment | added | Qiaochu Yuan | +1 for the bridges metaphor; I get that feeling a lot. | |
| Feb 9, 2010 at 11:19 | comment | added | Yemon Choi | Perhaps one other reason we like to stick topologies on things is to make our problem/framework "more finitary" (the liking for compactness-type properties on various spectra-of-algebras, for instance). | |
| Feb 9, 2010 at 11:07 | history | answered | Andrew Stacey | CC BY-SA 2.5 |