Timeline for More general metric spaces (where image of metric is not a subset of $\mathbb R$)
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 26, 2018 at 3:32 | comment | added | Anton Petrunin | Your construction reminds the the construction in the Hilber's 4th problem. The "distance" is the set of lines separating two points in the plane. In particular, a measure on the set of all lines then produces a metric. | |
| Aug 25, 2018 at 15:06 | comment | added | Henno Brandsma | I once attended a lecture on such a generalisation. It turns out that weakening the codomain to some structure like an Abelian semigroup with compatible order, say, we get that all Tychonoff spaces are "weakly metrisable" that way. So we get the uniformisable spaces. Cannot recall a reference, though. | |
| Aug 25, 2018 at 11:49 | comment | added | YCor | There is a notion of $\Lambda$-ree (en.wikipedia.org/wiki/…) where distances are valued in some totally ordered abelian group. Also distances valued in non-standard reals naturally naturally when considering ultraproducts of metric spaces. | |
| Aug 25, 2018 at 11:48 | comment | added | YCor | I don't know why bothering with the image of the distance map: instead discuss by considering the target set, which is $\mathbf{R}$. | |
| Aug 25, 2018 at 11:35 | answer | added | Gerald Edgar | timeline score: 3 | |
| Aug 25, 2018 at 11:17 | history | asked | Right | CC BY-SA 4.0 |