Timeline for Topological spaces determined by generalized metric spaces
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 15, 2012 at 15:50 | vote | accept | David Wasserman | ||
| Aug 14, 2012 at 18:37 | comment | added | Ramiro de la Vega | @Anton: I´m using the definition of "open" given by the OP´s link: $U$ is open iff for every $x \in U$ there is $r>0$ such that $B(x,r) \subseteq U$. With this definition it is obvious that the open sets form a topology. A different matter is whether sets of the form $B(x,r)$ are open or not. | |
| Aug 14, 2012 at 14:45 | comment | added | Anton Petrunin | @Ramiro, I guess you want to use open balls as prebase, but that is not "the same definition". | |
| Aug 14, 2012 at 13:33 | answer | added | Ramiro de la Vega | timeline score: 13 | |
| Aug 14, 2012 at 11:42 | comment | added | Ramiro de la Vega | @Anton: You don´t need anything to show that the open sets form a topology. It is just that the open balls might not be open so they are not a base for this topology. | |
| Aug 14, 2012 at 0:33 | answer | added | Mike Shulman | timeline score: 3 | |
| Aug 14, 2012 at 0:08 | answer | added | Nate Ackerman | timeline score: 15 | |
| Aug 13, 2012 at 23:56 | comment | added | Anton Petrunin | No, something IS used. In general the intersection of two "open balls" is not "open". | |
| Aug 13, 2012 at 22:00 | comment | added | Qiaochu Yuan | Careful: if $d$ isn't symmetric then there are two notions of open ball ("left open balls" and "right open balls") so you actually get two topologies. A nice example is the "counterclockwise distance" metric on $S^1$. One of the topologies you get is the "topology of counterclockwise convergence" and the other is the "topology of clockwise convergence." | |
| Aug 13, 2012 at 21:56 | history | asked | David Wasserman | CC BY-SA 3.0 |