Timeline for motivation for compactness [duplicate]
Current License: CC BY-SA 2.5
16 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 30, 2010 at 19:11 | comment | added | Alex Anderson | Hello, sorry that I reposted a question. Your comments have been very helpful! --Alex | |
| Jun 30, 2010 at 7:33 | comment | added | Pete L. Clark | I just noticed that the OP self-identifies as an undergraduate at Washington University in Saint Louis (whereas the phrasing of the question suggested to me that he was without university affiliation). In that case: you should ask these kinds of questions of the professors in the math department! That's what you're paying for (and, in part, what they're getting paid for). | |
| Jun 30, 2010 at 7:30 | comment | added | Pete L. Clark | The question has been closed. Four people voted for "exact duplicate"; I voted for "off topic". This is simply not a research level question: please see the FAQ for some appropriate websites. | |
| Jun 30, 2010 at 7:27 | history | edited | CommunityBot |
insert duplicate link
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| Jun 30, 2010 at 7:27 | history | closed |
Robin Chapman Harry Gindi Wadim Zudilin Greg Stevenson Pete L. Clark |
exact duplicate | |
| Jun 30, 2010 at 7:18 | comment | added | Yemon Choi | As Pietro says: although one can reel of a list of results where compactness is used in the proof, my own feeling is that one has to be patient when learning this kind of basic point-set topology, and try to find sources which discuss why compactness is necessary. Compactness starts to seem much more valuable once you see where things can go wrong in its absence. | |
| Jun 30, 2010 at 7:18 | comment | added | T.. | As this query heads for foreclosure, I'll add something not covered in the thread Robin cited. For visualing compactness there is the Heine-Borel theorem saying what it means for sets in ordinary geometric space, and in some sense that is that. However, this is misleading, in the sense that the real force of compactness comes not from intuition but from decades of accumulated experience; it is a tameness hypothesis (similar to measurable, separable, or Noetherian) that appears extremely often as a natural dividing line or organizing principle. | |
| Jun 30, 2010 at 7:12 | comment | added | KConrad | Your question is answered by Wikipedia, so look there: en.wikipedia.org/wiki/Compactness, which also explains how the concept developed. A quote from Frechet (who first defined metric spaces): "We have already pointed out and will recognize throughout this book the importance of compact sets. All those concerned with general analysis have seen that it is impossible to do without them." (For Frechet, compactness meant what we'd call sequential compactness, which for your analysis reading is equivalent to the open cover version.) | |
| Jun 30, 2010 at 7:11 | answer | added | supercooldave | timeline score: 1 | |
| Jun 30, 2010 at 7:08 | answer | added | Daniel Barter | timeline score: 0 | |
| Jun 30, 2010 at 6:47 | history | edited | Yemon Choi |
added some tags
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| Jun 30, 2010 at 6:42 | comment | added | Pietro Majer | Alex, doesn't your book report a number of theorems about compactness? Try finding counterexamples when the hypothesis of compactnes is dropped. | |
| Jun 30, 2010 at 6:26 | answer | added | Vlad K | timeline score: 2 | |
| Jun 30, 2010 at 6:12 | comment | added | Robin Chapman | You might consult the earlier question: mathoverflow.net/questions/25977 . | |
| Jun 30, 2010 at 5:55 | history | asked | Alex Anderson | CC BY-SA 2.5 |