Timeline for Non trivial vector bundle over non-paracompact contractible space
Current License: CC BY-SA 3.0
7 events
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| Sep 7, 2012 at 16:50 | comment | added | Todd Trimble | Well, $\mathbb{R}^J$ is certainly contractible: the map $(t; x_1, x_2, \ldots) \mapsto (t x_1, t x_2, \ldots)$ is continuous. The long line however is not contractible; see here: mathoverflow.net/questions/35087/… | |
| Sep 7, 2012 at 15:05 | comment | added | David White | It's been a long time since I actively studied point-set topology, but isn't $\mathbb{R}^J$ for some uncountable $J$ an example of a space which is not paracompact? It seems like it should be Hausdorff, since it's a product of Hausdorff spaces. It also feels contractible, but maybe it's not. Anyway, it's the best I could come up with for a potential base-space to handle this last example. Another I thought of was the order topology $S_\Omega$ (see Munkres), and I also feel like it's contractible because it's not too long yet to be reeled in. | |
| Sep 7, 2012 at 13:15 | comment | added | Ramón Barral | Thanks for your answer. This also deals with the non-paracompact non-Hausdorff case, using a suitable wedge sum with the trivial bundle over a non-paracompact contractible space, so only the Hausdorff non-paracompact case remains without an example. | |
| Sep 7, 2012 at 12:54 | vote | accept | Ramón Barral | ||
| Sep 7, 2012 at 12:23 | history | edited | Tom Goodwillie | CC BY-SA 3.0 |
added 59 characters in body
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| Sep 7, 2012 at 5:09 | comment | added | Todd Trimble | Did you mean to end in mid-sentence? | |
| Sep 7, 2012 at 1:47 | history | answered | Tom Goodwillie | CC BY-SA 3.0 |