Timeline for Classes of metric spaces with additional structure [closed]
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 11, 2011 at 12:24 | history | closed |
Yemon Choi Daniel Litt Andrés E. Caicedo Ryan Budney Simon Thomas |
off topic | |
| May 11, 2011 at 7:50 | comment | added | Pietro Majer | As an additional property, what about compatible group or vector space structures. For instance, a real topological vector space is metrizable if and only if it has a countable basis of nbd's of the origin. | |
| May 11, 2011 at 6:37 | comment | added | Marek | @Yemon: no, I don't want to learn only about Polish spaces (or rather Polish metric spaces), that was just one particular example that illustrates that certain topological property is required in some area. I wanted to get a list of similar examples to better understand which properties (or additional structures) are important and why, thereby better understanding classes of metric spaces. If such a list/classification doesn't exist (which might very well be possible) then that's just fine as an answer to my question. Thanks. | |
| May 11, 2011 at 6:32 | comment | added | Yemon Choi | Also, in response to your comment: we are trying to point you in the right direction, but you keep saying this is not what you are after... | |
| May 11, 2011 at 6:30 | comment | added | Yemon Choi | Marek: I guess one thing which confuses me is that what you say in the comments is not quite what you say in the body of the main question. Your main question seems to suggest you want to learn about Polish spaces, in which case the comments of Andreas and myself apply; but your comments seem to want a classification of metric spaces in the belief that such a thing will be useful to you. I am not aware of any such classification, and am not convinced it would be useful to you. Not everything in mathematics is easily classified, and this is why one has a general framework to unify examples | |
| May 11, 2011 at 6:13 | answer | added | Spice the Bird | timeline score: 1 | |
| May 11, 2011 at 5:42 | comment | added | Marek | @Andreas: I didn't miss this point at all and am not sure why you would say so. In any case, I think I have by now stated quite clearly what I am after and if no one intends to point me into right direction but instead only dwell on how I misunderstood this and forget to read that (which I didn't, in both cases), I might as well delete this question... Sorry for taking your time, I didn't realize the question would be so badly received. | |
| May 10, 2011 at 23:45 | comment | added | Andreas Blass | @Marek: Your comment "every complete separable metric space is Polish" suggests (to me) that you missed the point of Yemon's pervious comment. Although what you wrote is true, it's also true that some incomplete separable metric spaces are Polish (because the same topology can be induced by a different, complete metric). The property of being Polish is a matter of topology; even if you're given a topology as induced by a metric, you may need to change the metric (but not the topology) to see that your space is Polish. | |
| May 10, 2011 at 23:40 | comment | added | Yemon Choi | A POLISH SPACE IS NOT A METRIC SPACE. It is a separable topological space which can be equipped with a complete metric. There are many different metric spaces on $R^2$ which yield the same (usual) topology, and making a list of all of them will tell you very little of worth about the Polish space $R^2$ that is not already obvious from using the "usual" metric | |
| May 10, 2011 at 21:14 | comment | added | Marek | @Yemon: still, thanks for the effort. I'll try to look at the book you suggested. As I said, it is quite probably that my question doesn't need more than going through basic results. The hard part is figuring out where to find those :) | |
| May 10, 2011 at 21:12 | comment | added | Marek | @Yemon: sorry, if I didn't make myself clear but what I am really after is some classification of metric space, so I can make some sense of them. Classification seems to me a basic way to learn about some topic. What I am asking about is essentially those types and why are they useful. As for the Polish space... every complete separable metric space is Polish, and so is an important example of a metric space. The fact that such a basic example is missing suggests that other similarly basic examples are missing too and so the entry isn't terribly useful for my purposes. | |
| May 10, 2011 at 19:32 | comment | added | Yemon Choi | On looking at that wikipedia entry, I think that the examples given there are worth looking at. Probably the reason why you don't see a mention of Polish spaces is that they are topological spaces that happen to be separable and metrizable; they are not metric spaces per se, because different metrics can give the same topology. Unusurprisingly, Wikipedia's entry on Polish spaces en.wikipedia.org/wiki/Polish_space is a better place to look for those examples. | |
| May 10, 2011 at 19:28 | comment | added | Yemon Choi | Metric spaces are an extremely useful abstraction of structure found in many interesting examples. It is better to learn about these examples rather than ask "what can we add to the definition of a metric space to obtain interesting things"? | |
| May 10, 2011 at 19:26 | comment | added | Yemon Choi | I think your 1st question is too vague. One can always think up extra "bells and whistles" to add a given definition/concept; but that's not really how one should do mathematics, IMHO. If you want to learn about topological and metric spaces then a good introductory textbook (Sutherland's "Introduction to metric and topological spaces" is a bit basic, but might do) is worth far more than a list of cool-sounding things | |
| May 10, 2011 at 17:40 | comment | added | Marek | @Theo: I have read the entry on metric spaces and while it contains useful information I believe it doesn't really answer my question (e.g. there's no mention of the Polish space at all). As for the category page, I haven't seen that but it also doesn't seem to answer my question (again no mention of Polish spaces). Just throwing a glossary of some random terms that are somehow connected with metric spaces, you're not really helping, sorry. | |
| May 10, 2011 at 16:07 | comment | added | Theo Buehler | Honestly, have you read the wikipedia page on metric spaces before posting? It seems to me that more or less any reasonable property can be found there en.wikipedia.org/wiki/Metric_space and then by browsing through en.wikipedia.org/wiki/Category:Metric_geometry | |
| May 10, 2011 at 15:49 | history | asked | Marek | CC BY-SA 3.0 |