Timeline for Extending the topology on a set to the group/vector space it generates
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 14, 2016 at 20:48 | vote | accept | Mike Battaglia | ||
| Jul 7, 2016 at 17:17 | comment | added | Mike Battaglia | Thanks Karol - added a bounty to see if I can get an answer also addressing infinite linear combinations, but if not I'll accept this one. | |
| Jul 6, 2016 at 10:24 | comment | added | Karol Szumiło | If we take the discrete topology on $\mathbb{Z}$, then topological $\mathbb{Z}$-modules are exactly topological abelian groups so that seems like the most natural choice. This way my construction describes free topological abelian groups on topological spaces. | |
| Jul 5, 2016 at 0:49 | comment | added | Mike Battaglia | still hung up by the choice of topology on $\Bbb Z$. What I was really getting at was a "natural" canonical way to extend the topology on a set to the free abelian group on that set. However, the results of this construction seem to depend on the topology placed on $\Bbb Z$. Is there some topology on $\Bbb Z$ that should be singled out as "natural" for this purpose, or are there as many topologies on the free abelian group on a set as there are compactly generated topologies on $\Bbb Z$? | |
| Jun 29, 2016 at 21:56 | comment | added | Karol Szumiło | You can take any topology on $\mathbb{Z}$ that makes it into a (compactly generated) topological ring. | |
| Jun 29, 2016 at 21:56 | comment | added | Karol Szumiło | If $X$ is discrete of cardinality $\mathfrak{c}$, then it is obviously compactly generated. What I'm saying is that if you carry out the construction as I described in the category of all topological spaces, then $\mathbb{R} X$ will not be a TVS, but if you do it in the category of compactly generated spaces it will be a TVS. | |
| Jun 29, 2016 at 21:37 | comment | added | Mike Battaglia | Also, what topology are you putting on $\Bbb Z$ here - the discrete topology? The profinite topology? Something else? | |
| Jun 29, 2016 at 21:30 | comment | added | Mike Battaglia | Thank you Karol - this is a great point of departure. One question, though - what do you mean by discrete spaces of cardinality c not being compactly generated? Every discrete space is first-countable, and every first-countable space is compactly generated. | |
| Jun 29, 2016 at 12:00 | history | answered | Karol Szumiło | CC BY-SA 3.0 |