Timeline for What's wrong with compact-open topology on the space of maps?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| May 14, 2010 at 20:38 | comment | added | Sergei Ivanov | I just noted that the particular construction in my first comment is not countable. And cannot be, because it works in the non-metrizable case as well. | |
| May 14, 2010 at 19:51 | comment | added | Igor Belegradek | Not sure I understand the last comment. Andrew, does say that with $C^\infty$ topology $\gamma(E)$ is Fréchet. Indeed, consider semi-norms corresponding to an exhaustion by a countable family of compact sets. | |
| May 14, 2010 at 18:29 | comment | added | Sergei Ivanov | Not quite so, you need to be careful if you want a countable family of semi-norms, as explained in Andrew's answer. For local contractibility, this does not matter. | |
| May 14, 2010 at 13:59 | comment | added | Igor Belegradek | Looks like you are confirming that $\Gamma(E)$ is a Fréchet space. I thought this should be the case, but then I got confused. Thanks! | |
| May 14, 2010 at 13:36 | comment | added | Sergei Ivanov | For every local trivialization of $E$, every compact subset $K$ of the respective region in $M$ and every positive integer $r$ you define a semi-norm on $\Gamma(E)$ as the coordinate-wise $C^r$ norm of the restriction of a section to $K$. The compact-open $C^\infty$ topology is the topology defined by this family of semi-norms. Thus $\Gamma(E)$ is locally convex topological vector space and hence locally contractible. | |
| May 14, 2010 at 13:04 | comment | added | Igor Belegradek | I am interested in sections of vector bundles because they form a vector space, and hence, $\Gamma(E)$ is more likely to be locally contractible. As you say the space of maps $C^\infty(M, N)$ need not be locally contractible. One specific space I care about is the space of $(r,s)$-tensors on a non-compact manifold. | |
| May 14, 2010 at 9:55 | history | edited | Sergei Ivanov | CC BY-SA 2.5 |
added non-vector example
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| May 14, 2010 at 9:07 | history | answered | Sergei Ivanov | CC BY-SA 2.5 |