Timeline for Properly Discontinuous Action
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| S Jul 22, 2015 at 10:38 | history | suggested | CommunityBot | CC BY-SA 3.0 |
fix unnecessary escaping: "\\,:\\," introduces undesirable linebreaks. Minor change of wording: we have that -> we see that
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| Jul 22, 2015 at 10:25 | review | Suggested edits | |||
| S Jul 22, 2015 at 10:38 | |||||
| Feb 28, 2011 at 0:36 | comment | added | Theo Buehler | Anyway, thanks for pointing out that I didn't make myself perfectly clear. I'll try to clarify some things some time tomorrow. | |
| Feb 28, 2011 at 0:08 | comment | added | Theo Buehler | @ACL: Are you sure about the changes in terminology? I've never observed such a thing but I didn't look for it. Anyway in my edition it's defined as you say but that's equivalent to the definition I gave. Indeed, in Ch I. §10, No 2, Thm 1 they prove that for a continuous map $f \times \operatorname{id}_{Z}$ is closed for all $Z$ iff $f$ is closed and $f^{-1}(y)$ is quasi-compact. I hinted at that in my second remark above. In Prop. 6 they prove that $f^{-1}(K)$ is quasi-compact for all quasi-compact $K$ without further hypotheses. Since points are quasi-compact, the two definitions coincide. | |
| Feb 28, 2011 at 0:07 | comment | added | ACL | Addendum: This second definition ($f$ universally closed) is equivalent to the one in your second remark : $f$ closed with compact fibres. | |
| Feb 27, 2011 at 23:39 | comment | added | ACL | Unfortunately, Bourbaki's terminology differs from edition to edition, and from French to English. So in the newest French edition, a map $f:X\to Y$ is said to be proper if the map $f\times{\rm Id}_Z:X\times Z\to Y\times Z$ is closed. The notion coincides with the one discussed above when $X$ is Hausdorff and $Y$ is Hausdorff and locally compact. (It implies that $X$ is locally compact.) | |
| Feb 24, 2011 at 6:10 | history | answered | Theo Buehler | CC BY-SA 2.5 |