Timeline for Is there a topological space X homeomorphic to the space of continuous functions from X to [0, 1]?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 22, 2017 at 8:02 | vote | accept | Qiaochu Yuan | ||
| Mar 22, 2017 at 7:50 | answer | added | Sam Eisenstat | timeline score: 33 | |
| Mar 17, 2017 at 22:37 | comment | added | Qiaochu Yuan | Oh, interesting. For some reason I was under the impression that either the exponential topology agreed with the compact-open topology or the exponential didn't exist. In that case I of course want the exponential topology. | |
| Mar 17, 2017 at 12:18 | comment | added | Todd Trimble | I'd pause before decreeing it should be the compact-open topology; I'd say it should be the exponential topology first and foremost, i.e., the topological structure such that there is a natural isomorphism $Top(-, [X, Y]) \cong Top(- \times X, Y)$. This need not be the compact-open topology (although it will be if $X$ is say locally compact Hausdorff). | |
| Mar 17, 2017 at 5:52 | comment | added | მამუკა ჯიბლაძე | There are two candidates one might consider as a first approximation: the direct limit of $C^k(I)\to C^{k+2}(I)\to C^{k+4}(I)\to...$ or the inverse limit of $...\to C^{k+4}(I)\to C^{k+2}(I)\to C^k(I)$ (the latter is $C(\_)$ of the former) | |
| Mar 17, 2017 at 5:22 | comment | added | Qiaochu Yuan | Yes, the compact open topology. If I'm allowed to pick the topology on $[X, Y]$ then lots of silly things could happen, I imagine. | |
| Mar 17, 2017 at 5:20 | comment | added | Cusp | I guess you are considering the compact open topology on $[X,Y]$? Or is the question is also about finding the topology on $[X,Y]$? | |
| Mar 17, 2017 at 2:36 | comment | added | Qiaochu Yuan | Yes, I suppose that's worth clarifying. Edited. | |
| Mar 17, 2017 at 2:35 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
deleted 184 characters in body; edited title
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| Mar 17, 2017 at 2:06 | comment | added | Igor Rivin | Functions = continuous functions, I assume? | |
| Mar 17, 2017 at 2:01 | history | asked | Qiaochu Yuan | CC BY-SA 3.0 |