Timeline for Image of the Hilbert space under a continuous bijection
Current License: CC BY-SA 4.0
12 events
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| Apr 17, 2023 at 12:17 | history | edited | Igor Belegradek | CC BY-SA 4.0 |
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| Feb 13, 2013 at 16:49 | answer | added | Ramiro de la Vega | timeline score: 1 | |
| Oct 8, 2011 at 4:23 | comment | added | Will Sawin | The bridge isn't actually made of points, it just reduces the distance. | |
| Oct 7, 2011 at 14:29 | comment | added | Igor Belegradek | @Will Sawin: your "bridge example" cannot work as $Y$ because after removing two points on the bridge the space becomes disconnected. | |
| Oct 7, 2011 at 12:10 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| Oct 7, 2011 at 3:52 | answer | added | Will Sawin | timeline score: 2 | |
| Oct 7, 2011 at 3:19 | comment | added | Will Sawin | In an infinite-dimensional Hilbert space, since it's not even locally compact, you can do all sorts of weird stuff. Choose a sequence $a_1,a_n,..$ with no limit points and decide on a point $a$ you want it to converge to. Put a bridge of length $1/2^n$ from $a_n$ to $a$, and $a$ becomes the limit of that sequence. You can do this a lot of times, as long as you make sure that you never reduce the distance between two points to $0$. I don't think the separability is necessary. The image of a countable dense subset of $X$ is a countable dense subset of $Y$. | |
| Oct 7, 2011 at 3:09 | comment | added | Will Sawin | A continuous bijection $X$ to $Y$ is just a coarsening of the topology of $X$. There are a lot of coarsenings of that topology. Metrizable coarsenings just correspond to metrics that are continuous in both variables. Obviously, the 6-thing, in which far away points become very close, can happen. I'm actually shocked by my inability to find other examples of bad behavior. The only reason I can imagine that there wouldn't be anything else is if every continuous metric on a compact metrizable space gives the same topology. | |
| Oct 7, 2011 at 1:03 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| Oct 7, 2011 at 1:01 | comment | added | Igor Belegradek | @Bill: thanks! I obviously misapplied invariance of domain. | |
| Oct 7, 2011 at 0:37 | comment | added | Bill Johnson | I do not understand (3). The real line bijects continuously onto the figure 6 minus its top point. | |
| Oct 6, 2011 at 23:54 | history | asked | Igor Belegradek | CC BY-SA 3.0 |