Timeline for Does a uniform space have a closed embedding in a product of metric spaces?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 9, 2012 at 21:06 | comment | added | jbc | Sorry, I had misread the question to mean embeddable as a closed subset of a product of complete metric spaces. | |
| Dec 9, 2012 at 20:53 | vote | accept | Michael Barr | ||
| Dec 9, 2012 at 20:51 | comment | added | Michael Barr | No the answer is not "complete". It is clear that complete uniform spaces are the limit closure of the complete metric spaces, a question I had already answered. As for "realcompact" the uniform spaces whose associated topology is realcompact appear to be the limit closure of the full subcategory whose only object is the reals. I have not yet written up the details of that, so I am not quite certain. | |
| Dec 9, 2012 at 19:30 | comment | added | Igor Khavkine | @jbc: There are plenty of incomplete metric spaces, yet each is also a uniform space and a closed subspace of itself (a product space with one factor). Seems to me that the condition of the question is distinct from completeness. | |
| Dec 9, 2012 at 18:59 | comment | added | jbc | The condition of the question is equivalent to the uniform space being complete. The other concepts mentioned here are topological and so not appropriate in this context. One could, of course, pose the same question for completely regular spaces with, e.g., the finest uniformity compatible with the topology and so obtain results of this type. | |
| Dec 9, 2012 at 18:36 | answer | added | Ramiro de la Vega | timeline score: 5 | |
| Dec 9, 2012 at 18:14 | comment | added | Todd Eisworth | Is the notion of "realcompact" what you are looking for? | |
| Dec 9, 2012 at 18:06 | comment | added | Michael Barr | Sorry, I got mixed up between semi- and pseudo-metric. Having just googled them, I see I should have said pseudo-metric. You are right, the question was senseless. But I still would like to know the answer. Maybe I should mention the context. I can classify the limit closure, in the category of uniform spaces, of the metric spaces as those uniform spaces that are closed subspaces of a product of metric spaces and want to know if that is all uniform spaces. If only John Isbell were still around to ask. | |
| Dec 9, 2012 at 17:39 | comment | added | Igor Khavkine | Having some trouble parsing the question. Likely because of a () mismatch. And do you mean semi-metric or pseudo-metric? As defined here, for instance: en.wikipedia.org/wiki/Uniformity_%28topology%29 | |
| Dec 9, 2012 at 17:05 | history | asked | Michael Barr | CC BY-SA 3.0 |