Timeline for Meaning of "Compact" in 1932 Paper by van der Waerden "Continuity Theorem for Semisimple Lie Groups".
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| May 11, 2011 at 13:31 | vote | accept | Selene Routley | ||
| May 9, 2011 at 14:21 | comment | added | Gerald Edgar | 1932? Wasn't it Alexandroff & Hopf 1935 who propose bicompact="every open cover has a finite subcover" as the primary definition, rather than the previous meaning compact="every sequence has a convergent subsequence"? Only after non-metric spaces were frequently used did the difference became apparent. | |
| May 9, 2011 at 4:48 | comment | added | Pete L. Clark | (Note that, as Qiaochu suggests in his answer, it is hard to believe that van der Waerden was actually entertaining metaplectic groups in his paper. But it would be interesting to know for sure...) | |
| May 9, 2011 at 4:42 | comment | added | Pete L. Clark | @LSpice: Hmm. To me semisimple groups are necessarily linear, so the metaplectic groups do not qualify. But maybe that's just me...Can we get a ruling on this one? I'm starting to think I should quit while I'm behind. | |
| May 9, 2011 at 4:04 | comment | added | LSpice | P L Clark, maybe I've got my terminology mixed up; but isn't, say, $\operatorname{Mp}(2, \mathbb R)$ a semisimple Lie group that is not, or at least not par naissance, a subgroup of a general linear group? | |
| May 9, 2011 at 3:42 | comment | added | Selene Routley | @Pete: " by a "manifold" most people meant things that could a priori be embedded into Euclidean space" - is an interesting piece of history in itself and could pretty much answer my question, if a someone with history background could substantiate it. Certainly your comment is highly plausible - it couldn't be much otherwise and besides why would Whitney look for a theorem in that direction if it weren't something near to what many people were thinking. | |
| May 9, 2011 at 3:19 | comment | added | Pete L. Clark | @Mariano: actually, I just took a look via amazon.com at the latest (English) edition of "General Topology", and I still see quasi-compact in the table of contents. | |
| May 9, 2011 at 3:17 | answer | added | Qiaochu Yuan | timeline score: 2 | |
| May 9, 2011 at 3:17 | comment | added | Pete L. Clark | @Qiaochu: thanks for that -- Whitney embedding theorems were proved circa 1936. But this was not exactly my point: before Whitney embedding, by a "manifold" most people meant things that could a priori be embedded into Euclidean space. This is certainly true for a semisimple Lie group, since it is a subgroup of a general linear group. Right? | |
| May 9, 2011 at 3:05 | comment | added | Qiaochu Yuan | Peter-Weyl was proven in 1927. | |
| May 9, 2011 at 2:47 | comment | added | Mariano Suárez-Álvarez | Pete, aren't newer editions of Topologie Générale de-quasi-fied? | |
| May 9, 2011 at 2:32 | comment | added | Selene Routley | Hello Pete. Many thanks for your comments. As for your first one, I was under the impression that the Whitney embedding theorems were not around in 1932 - I guess though that they were pretty strongly believed conjectures though at the time? (See Wikipedia page en.wikipedia.org/wiki/History_of_manifolds_and_varieties). | |
| May 9, 2011 at 2:12 | comment | added | Pete L. Clark | By the way, so far as I know, in 1932 a compact space was a topological space in which every open cover has a finite subcover. In other words, the same as today modulo the Bourbakistes who sometimes use "quasi-compact" for this (including me!). But anyway we are talking about Hausdorff, even metrizable spaces... | |
| May 9, 2011 at 2:08 | comment | added | Pete L. Clark | Every Lie group is in particular a manifold so can be embedded in Euclidean space. So a compact Lie group is homeomorphic to a compact subset of Euclidean space, thus Bolzano-Weierstrass applies. Is there more to it than this? | |
| May 9, 2011 at 2:06 | history | edited | Selene Routley | CC BY-SA 3.0 |
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| May 9, 2011 at 2:00 | history | edited | Selene Routley | CC BY-SA 3.0 |
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| May 9, 2011 at 1:55 | history | asked | Selene Routley | CC BY-SA 3.0 |