Timeline for Why is a topology made up of 'open' sets? [closed]
Current License: CC BY-SA 4.0
84 events
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| Nov 11 at 23:04 | comment | added | Red Banana | @KConrad Maybe the real open sets were the clopen sets we defined along the way? | |
| Mar 20 at 15:33 | comment | added | Michael Hardy | One of the more prominent among frequently occurring intellectual sins of mathematicians is to omit to explain things like this. If you've ever taught a linear algebra course in which the definition of matrix multiplication is presented dogmatically rather than by saying the reason it's done that way is to make matrix multiplication correspond to composition of linear transformations, then you are guilty. I don't recall seeing a textbook introducing this concept without committing this sin. Quite possibly some such innocent book exists. (?) | |
| Jul 3, 2023 at 17:07 | history | left closed in review |
Alex M. Daniele Tampieri Alexey Ustinov |
Original close reason(s) were not resolved | |
| Jul 3, 2023 at 3:11 | review | Reopen votes | |||
| Jul 3, 2023 at 17:07 | |||||
| Nov 17, 2019 at 17:55 | history | edited | J. W. Tanner | CC BY-SA 4.0 |
corrected spelling of measurement
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| Aug 25, 2019 at 15:45 | comment | added | William Oliver | @DeaneYang The definition of a topological space is sparse, true, but so is the definition of a monoid. When you think about it, the definition of the topological space is very close to that of a monoid. The only difference is that there are two binary operations (union, and intersection) and two identity elements (empty set and the universe) instead of the usual one. And they are both commutative. IMO, if it is intuitive that a monoid should be useful when working with discrete objects, it should also be intuitive that a topological space would be as well. | |
| Apr 8, 2019 at 4:40 | review | Reopen votes | |||
| Apr 9, 2019 at 1:37 | |||||
| Aug 9, 2018 at 12:41 | comment | added | FWE | U might want to have a look at this historical explanation at mathstack math.stackexchange.com/questions/904983/… | |
| Dec 30, 2017 at 23:45 | review | Reopen votes | |||
| Jan 4, 2018 at 9:05 | |||||
| Dec 1, 2017 at 16:01 | comment | added | MarkWayne | Helpful, relevant discussion that takes no resources away from the site in the large. Stupidly closed, yet again. | |
| Nov 30, 2017 at 4:40 | review | Reopen votes | |||
| Dec 3, 2017 at 11:49 | |||||
| Oct 26, 2017 at 3:04 | review | Reopen votes | |||
| Oct 28, 2017 at 11:29 | |||||
| Oct 19, 2017 at 2:58 | review | Reopen votes | |||
| Oct 19, 2017 at 11:12 | |||||
| Nov 25, 2016 at 12:24 | history | edited | Gerry Myerson |
edited tags
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| Nov 25, 2016 at 7:15 | comment | added | Duchamp Gérard H. E. | @HarryGindi [Nets are the poor man's filters =)!]----> by no means : nets are brilliant for proofs with limits and filters are enlightning when domains are concerned (like germs for instance). I use both ... | |
| Nov 25, 2016 at 6:53 | history | edited | Martin Sleziak |
added some tags (The post has been bumped anyway by an edit to an answer.)
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| Oct 6, 2015 at 16:28 | comment | added | Remember me | It works the same way as categories . To get their motivation you need to actually see categories | |
| Jun 18, 2014 at 21:38 | comment | added | KConrad | It's ironic that this question about open sets is now closed. | |
| Dec 7, 2013 at 0:27 | comment | added | Steven Gubkin | I would like to point out that sometimes other topologies arise naturally when thinking about concrete problems. For example, consider a function $f:\mathbb{R} \to \mathbb{R}$ which gives the strength of a beam given its radius. For some engineering goals it might only matter that the strength is greater than or equal to some given value. So put topology of half open rays on the codomain. What are the continuous functions? Why does it make intuitive sense in this situation? Jumps may be physically possible because you need a new material at a certain radius. | |
| Sep 10, 2013 at 13:01 | review | Reopen votes | |||
| Sep 10, 2013 at 13:04 | |||||
| Sep 9, 2013 at 12:58 | review | Reopen votes | |||
| Sep 9, 2013 at 13:05 | |||||
| Jan 16, 2012 at 5:00 | history | closed |
Dan Petersen Felipe Voloch user6976 Bill Johnson Andy Putman |
no longer relevant | |
| Jan 14, 2012 at 14:30 | answer | added | Ronnie Brown | timeline score: 7 | |
| Jan 14, 2012 at 7:11 | answer | added | Toby Bartels | timeline score: 17 | |
| Jan 9, 2012 at 5:55 | answer | added | Matt Brin | timeline score: 3 | |
| Dec 3, 2011 at 9:48 | answer | added | Will Sawin | timeline score: 3 | |
| Dec 3, 2011 at 9:31 | comment | added | Will Sawin | Simplicial complexes and other elements of combinatorial geometry, as you define it, can be expressed as finite topological spaces. Applying an ordinary topological cohomology theory like Cech then gives the correct answer. | |
| Dec 3, 2011 at 8:57 | answer | added | Michael | timeline score: 7 | |
| S Jun 27, 2011 at 14:15 | answer | added | Giorgio Mossa | timeline score: 6 | |
| S Jun 27, 2011 at 14:15 | history | made wiki | Post Made Community Wiki | ||
| Jun 26, 2011 at 23:07 | answer | added | Steve Hurder | timeline score: 26 | |
| Dec 14, 2010 at 21:34 | answer | added | Mark Bennet | timeline score: 14 | |
| Dec 14, 2010 at 21:13 | answer | added | David Feldman | timeline score: 24 | |
| Nov 24, 2010 at 1:44 | answer | added | Sándor Kovács | timeline score: 4 | |
| Nov 24, 2010 at 0:10 | answer | added | Michael Hardy | timeline score: 2 | |
| Jul 2, 2010 at 7:40 | answer | added | Elemer E Rosinger | timeline score: 7 | |
| Jul 2, 2010 at 1:42 | answer | added | T.. | timeline score: 11 | |
| Jul 1, 2010 at 22:50 | answer | added | Jonathan Wise | timeline score: 8 | |
| Jul 1, 2010 at 20:40 | answer | added | Terry Tao | timeline score: 244 | |
| May 16, 2010 at 12:39 | answer | added | David Wheeler | timeline score: 9 | |
| Mar 29, 2010 at 5:26 | answer | added | Sean Tilson | timeline score: 3 | |
| Mar 27, 2010 at 17:58 | answer | added | Marcos Cossarini | timeline score: 23 | |
| Mar 26, 2010 at 22:59 | comment | added | Deane Yang | Minhyong, if you really are talking about undergraduates, then I think all of the answers that refer to a metrizable topology (e.g., in analysis or differential geometry and topology) are appropriate. But it appears to me that you are talking about the use of topology in other more abstract settings, where as I said above, it is all a total mystery to me. | |
| Mar 26, 2010 at 16:51 | answer | added | Agustí Roig | timeline score: 2 | |
| Mar 26, 2010 at 5:43 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Mar 26, 2010 at 2:49 | answer | added | Justin Hilburn | timeline score: 39 | |
| Mar 25, 2010 at 23:45 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Mar 25, 2010 at 23:38 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Mar 25, 2010 at 1:46 | answer | added | Andrew Stacey | timeline score: 161 | |
| Mar 24, 2010 at 13:30 | answer | added | Donu Arapura | timeline score: 9 | |
| Mar 24, 2010 at 13:29 | comment | added | Deane Yang | Regarding the effectiveness of the standard definition of "topology": I feel comfortable with its effectiveness in areas such as functional analysis and differential geometry. But I have never understood why the standard definitions of topology should be useful at all when working with finite fields and other discrete objects. Is there any way to motivate that for the non-expert? | |
| Mar 24, 2010 at 13:16 | comment | added | Harry Gindi | Filters are equivalent to nets in a topological context, Andrew L. The difference is that filters also have application in logic and set theory as well. | |
| Mar 24, 2010 at 12:29 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Mar 24, 2010 at 12:17 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Mar 24, 2010 at 12:10 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Mar 24, 2010 at 10:03 | answer | added | gowers | timeline score: 33 | |
| Mar 24, 2010 at 8:50 | answer | added | Qiaochu Yuan | timeline score: 25 | |
| Mar 24, 2010 at 7:40 | comment | added | The Mathemagician | Everyone should read the late Robert Bartle's wonderful classic paper on generalized convergence,it's still the definitive source on the topic. Filters are more general,but that generality comes at a price,they aren't as well-behaved as nets,which are very straightforward generalizations of sequences and thier properties follow very naturally by extension from sequences in first countable spaces. Sequences need first countable topological spaces to converge-like metric spaces-since convergence requires "most" points in the range of a sequence to "land" in an basic open set. | |
| Mar 24, 2010 at 5:07 | comment | added | Harry Gindi | Nets and filters are equally powerful to track convergence in topological spaces. However, it turns out that filters ae a much more useful and powerful notion in general, because they are a structure that can be put on any poset. Every net generates a filter, but not vice-versa. | |
| Mar 24, 2010 at 3:53 | answer | added | JuanOS | timeline score: 1 | |
| Mar 24, 2010 at 3:42 | answer | added | Vectornaut | timeline score: 158 | |
| Mar 24, 2010 at 3:08 | comment | added | faridrb | filters and nets are equally powerful logically, nets are useful because you can use your intuition of sequences in metric spaces (more or less), filters are useful because the statements about convergence become much shorter and prettier.. | |
| Mar 24, 2010 at 2:19 | comment | added | Harry Gindi | (And filters are precisely the things used to define a neighborhood base. They're much more powerful and general than nets.) | |
| Mar 24, 2010 at 2:17 | comment | added | Harry Gindi | Nets are the poor man's filters =)! | |
| Mar 24, 2010 at 2:01 | comment | added | Theo Johnson-Freyd | My favorite definition of a topological space is that it knows which sequences converge, and to what. Actually, as you can see by thinking about spaces with too many points, it's not good enough to use sequences to probe to probe the topology of your space, but nets are good enough: a topological space is equivalent to a set and some (coherent!) rules to determine which nets converge where. | |
| Mar 24, 2010 at 0:48 | comment | added | Harry Gindi | I believe that this is similar to what Qiaochu was talking about. | |
| Mar 24, 2010 at 0:42 | comment | added | Harry Gindi | Here's a paper that I found interesting a while ago. It gives a slightly weaker notion of a topological space using formal neighborhoods and formal "convergents". bioinf.uni-leipzig.de/~studla/Publications/PREPRINTS/… | |
| Mar 24, 2010 at 0:15 | comment | added | Qiaochu Yuan | I don't think so. One can think of the closure operator as measuring a generalized kind of convergence or a generalized kind of distance, as per sigfpe's answer. Closure operators show up naturally in many branches of mathematics - for example any relation between two sets gives rise to a closure operator on each set - and in my opinion they can be thought of on their own terms. | |
| Mar 24, 2010 at 0:12 | answer | added | Michael Benfield | timeline score: 8 | |
| Mar 24, 2010 at 0:00 | comment | added | Minhyong Kim | Qiaochu: I agree. But those axioms appear to me not quite as 'primitive' as the ones applying to open sets. In fact, they seem to presuppose some notion of openness or closedness as motivation. | |
| Mar 23, 2010 at 23:50 | answer | added | David Carchedi | timeline score: 7 | |
| Mar 23, 2010 at 23:36 | answer | added | Tim Perutz | timeline score: 11 | |
| Mar 23, 2010 at 23:36 | comment | added | Gerry Myerson | Maybe "environments" is what some of us call "neighborhoods" (and others of us call "neighbourhoods"). | |
| Mar 23, 2010 at 23:22 | comment | added | The Mathemagician | I'm like,say what on "environments"? I've discussed topology with both James Stasheff and John Terrilla,both pretty emienent topologists and neither one ever mentioned "environments"! | |
| Mar 23, 2010 at 23:16 | answer | added | Charlie Frohman | timeline score: 8 | |
| Mar 23, 2010 at 23:15 | answer | added | Deane Yang | timeline score: 54 | |
| Mar 23, 2010 at 23:14 | answer | added | M.G. | timeline score: 2 | |
| Mar 23, 2010 at 23:14 | comment | added | Charles Rezk | @Jan: can you give a reference for "environments"? I can't find anything about it. | |
| Mar 23, 2010 at 23:14 | comment | added | Qiaochu Yuan | Do you find the Kuratowski closure axioms intuitive? If so, then the proof of equivalence between the Kuratowski closure axioms and the standard axioms is not hard. | |
| Mar 23, 2010 at 23:08 | comment | added | Ryan Budney | You can also use the formalism of closed sets, or neighbourhoods. The neighbourhood formalism is one of the more analogous to metric space formalism. But the "reason" why people use open sets is that it simplifies all the quantifiers, in a sense because they're "hidden" in the definition of the sets you're using. | |
| Mar 23, 2010 at 23:04 | answer | added | Dan Piponi | timeline score: 300 | |
| Mar 23, 2010 at 23:03 | answer | added | Harald Hanche-Olsen | timeline score: 2 | |
| Mar 23, 2010 at 22:36 | comment | added | Jan Weidner | You can not only define a topological space by open sets but also by environments. The definition using environments is far easier to motivate and more intuitive in my opinion. | |
| Mar 23, 2010 at 22:25 | history | asked | Minhyong Kim | CC BY-SA 2.5 |