Timeline for Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)
Current License: CC BY-SA 2.5
9 events
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| Feb 9, 2010 at 7:19 | comment | added | Tim Porter | @Yemon and Tom: I agree with there being (at least) two ways in which spaces seem to come up. They are not disjoint, of course, and the intersection includes many of the spaces studied by Borsuk in Shape Theory. The 'open covering to nerve to simplicial complex' of Cech theory is used in both for organising the patches. The embedding theorems used by Borsuk (Compact metric spaces embedd into the Hilbert Cube) then give a nice link to Ilya's third class of spaces. I also like finite spaces, and spaces presented as posets, which do not quite fit into Tom's classes, or do they? | |
| Feb 9, 2010 at 7:15 | comment | added | Anonymous | There's also the topology on the integers in Fürstenberg's proof of the infinitude of primes. | |
| Feb 9, 2010 at 6:53 | comment | added | Tom Leinster | Yemon, I see what you mean. It's an interesting point. Maybe it's worth adding the two dual points of views on manifolds: (i) as colimits (direct limits) of coordinate patches; (ii) as limits (inverse limits) corresponding to equations. By (ii) I mean that when you write something like {(x, y) in R x R: x^2 + y^2 = 1}, you're defining the curve as the equalizer of a pair of maps from the product R x R to R, and products and equalizers are types of limit. Maybe that's relevant. | |
| Feb 9, 2010 at 6:31 | history | edited | Tom Leinster | CC BY-SA 2.5 |
added pointer to comment
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| Feb 9, 2010 at 4:55 | comment | added | Yemon Choi | Tom, this is probably hopelessly muddle-headed of me, but it struck me that some of your geometric examples have a "built up from simpler pieces feel", which could be "direct-limit-ish", and some of your spectrummy example have an "inverse-limit-ish" feel. Perhaps the latter, which tend to arise via contravariant functors, are indicating that it's the algebraic objects which are built out of simpler pieces; with colimits being taken to limits? | |
| Feb 9, 2010 at 4:11 | comment | added | Tom Leinster | Ah, good point. For a start, Banach spaces and Hilbert spaces are all around. | |
| Feb 9, 2010 at 3:42 | comment | added | Ilya Grigoriev | There is at least one additional kind: functional analysis topologies, especially all the various weak convergence topologies. | |
| Feb 9, 2010 at 0:30 | comment | added | François G. Dorais | +1+ε (The little extra is for the delightful term spectrummy.) | |
| Feb 9, 2010 at 0:21 | history | answered | Tom Leinster | CC BY-SA 2.5 |