Timeline for What spaces can be obtained from $\mathbb{R}^{n}$ by taking quotient spaces and subspaces?
Current License: CC BY-SA 2.5
10 events
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| Feb 3, 2018 at 0:17 | answer | added | Taras Banakh | timeline score: 2 | |
| Jan 30, 2018 at 19:19 | comment | added | Alec Rhea | You may have better luck with your motivating intuition if you look at all quotients and subspaces of the Surreal numbers (under an appropriate topology or collection of $\xi$-topologies). Certainly we can get topologies of any cardinality (in particular ones that are not second countable), but it would take a much more trained eye than mine to see if all topological spaces can be obtained this way. | |
| Jan 30, 2018 at 17:20 | answer | added | Will Brian | timeline score: 9 | |
| Feb 4, 2014 at 17:38 | comment | added | Gerrit Begher | Concerning your friends question: Take a look at Bill Lawvere's answer in mathoverflow.net/questions/127841/… "My paper about Volterra's functionals [...] discusses the unsuitability for functional analysis (as well as for homotopy theory) of the attempt to characterize continuity or cohesion using open sets or other contravariant structure." | |
| Feb 4, 2014 at 17:11 | answer | added | Hannes Thiel | timeline score: 11 | |
| Mar 8, 2011 at 21:17 | comment | added | Stephen S |
@Anton Petrunin: Finite dimension can't be right, since every compact metrizable space is a quotient of the Cantor set, and that includes things like $[0,1]^{\aleph_0}$.
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| Mar 8, 2011 at 15:48 | comment | added | Anton Petrunin | This way you can get only separable spaces with finite dimension and I guess you can get all of them (?) | |
| Mar 8, 2011 at 11:25 | comment | added | Martin Sleziak | If you used quotients and topological sums, you would get sequential spaces. Using subspaces, quotiens and sums, you would get subsequential spaces. (S. P. Franklin, M. Rajagopalan: On subsequential spaces, Topology. and its Applications 35 (1990), 1–19) Your class will definiely be a subclass of the class of subsequential spaces. I am not sure about the precise characterization. | |
| Mar 8, 2011 at 10:40 | comment | added | Qiaochu Yuan | Your motivation question seems much more interesting! Related questions have been discussed on MO before, e.g. mathoverflow.net/questions/19152/… . The answers that most convinced me were the ones involving logic and computability. They suggest that the definition of a topology is useful because it is absurdly general, hence general enough to include nice things. But it is not necessarily geometrically natural. I think Grothendieck once expressed an opinion that the definition is "wrong" e.g. for homotopy theory? | |
| Mar 8, 2011 at 10:25 | history | asked | Sam Nolen | CC BY-SA 2.5 |