Timeline for Smooth manifolds that don't admit a partition of unity
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Mar 29, 2010 at 21:48 | comment | added | David Carchedi | By the way, does paracompact WITHOUT Hausdorff imply the existance of partitions of unity? I feel like this might be a dumb question... | |
| Mar 25, 2010 at 10:51 | vote | accept | Aston Smythe | ||
| Mar 25, 2010 at 0:17 | comment | added | David Carchedi | More generally, the etale space of any sheaf over a manifold is always locally Euclidean but rarely Hausdorff. | |
| Mar 24, 2010 at 22:54 | comment | added | babubba | Do Banach/Hilbert manifolds admit partitions of unity? | |
| Mar 24, 2010 at 21:35 | answer | added | Pete L. Clark | timeline score: 13 | |
| Mar 24, 2010 at 18:43 | answer | added | Chris Schommer-Pries | timeline score: 10 | |
| Mar 24, 2010 at 18:34 | comment | added | David E Speyer | Hausdorff does not follow from any local condition. Consider the "line with a double point": two copies of $\mathbb{R}$ glued to each other along $\mathbb{R} \setminus \{ 0 \}$. This is locally as nice as you could want in any sense. | |
| Mar 24, 2010 at 18:33 | comment | added | Harald Hanche-Olsen | Yes, you need to specify Hausdorff. Consider the real line with an extra zero grafted onto it. It's locally Euclidean, but not Hausdorff. | |
| Mar 24, 2010 at 18:29 | comment | added | Aston Smythe | I mean manifolds that aren't second countable but are locally euclidean and Hausdorff .... do we need to specify Hausdorff here, surely it should from the manifold being locally euclidean. | |
| Mar 24, 2010 at 18:12 | comment | added | Charlie Frohman | If your manifolds are second countable, Hausdorff and locally Euclidian, then they are paracompact, and hence any open cover admits a partition of unity subordinate to it, at least in the smooth category. In the analytic category, you will not have partitions of unity, but that is because there are no bump functions. | |
| Mar 24, 2010 at 18:07 | history | asked | Aston Smythe | CC BY-SA 2.5 |