25
$\begingroup$

The modern definition of topology is 'a family of subsets of a set $X$ containing the empty set and $X$, closed under unions and finite intersections'.

In Grundzüge der Mengenlehre (1914) Hausdorff presented his set of four axioms for topological space that has undoubtedly influenced the modern definition, since they both emphasize the notion of open set. But who introduced the modern definition for the first time?

Hausdorff's axioms or Umgebungsaxiome (page 213 in Grundzüge der Mengenlehre):

(A) Jedem Punkt $x$ entspricht mindestens eine Umgebung $U_x$; jede Umgebung $U_x$ enthält den Punkt $x$.

(B) Sind $U_x$, $V_x$ zwei Umgebungen desselben Punktes $x$, so gibt es eine Umgebung $W_x$, die Teilmenge von beiden ist.

(C) Liegt der Punkt $y$ in $U_x$, so gibt es eine Umgebung $U_y$, die Teilmenge von $U_x$ ist.

(D) Für zwei verschiedene Punkte $x$, $y$ gibt es zwei Umgebungen $U_x$, $U_y$ ohne gemeinsame Punkt.

  • 0
    According to [Wikipedia](http://en.wikipedia.org/wiki/Topology#History) it was Kuratowski in 1922.2011-10-06
  • 1
    Hm. This is *very* strange. Kuratowski wrote down his closure axioms in [Sur l'opération $\overline{A}$ de l'Analysis Situs](http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3121.pdf) but he doesn't prove that these axioms are equivalent to the modern ones and doesn't propose that these axioms be taken as definition of a topological space. Both on the English and German Wikipedia biographies of Kuratowski claim that these axioms were introduced in *[Sur la notion d'ensemble fini](http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm1117.pdf)* which is manifestly nonsense, as this is about *finite sets*.2011-10-06
  • 6
    It seems to me that [this paper](http://mcs.cankaya.edu.tr/~kenan/Moore2008.pdf) gives a full-fledged and authorative answer to your question. The official reference is: Gregory H. Moore, *[The emergence of open sets, closed sets, and limit points in analysis and topology](http://dx.doi.org/10.1016/j.hm.2008.01.001)*, Historia Mathematica **35** (2008) 220–241.2011-10-06
  • 0
    @t.b.: Indeed it gives the answer and much more. Quite interesting stuff. Thank you very much! According to the article, in 1925 Aleksandrov gave these two axioms in an article in Mathematische Annalen: (1) the intersection of two open sets is open, and the union of any set of open sets is open; (2) any two distinct points are contained in disjoint open sets. As the article notes, dropping the axiom (2) we almost get the modern definition.2011-10-06

2 Answers 2

25

A rather detailed and interesting discussion of the extremely convoluted history can be found in the paper by Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220–241.


It seems fair, if overly simplistic, to say that after Hausdorff, the following works were the main contributions towards the modern axiomatisation of topology:


Added: Bourbaki (who else?) pushed towards the modern accepted version and credit should also be given to Kelley's classic topology book General topology. See Moore's paper mentioned at the beginning for more details on this, especially section 14.


Added later: For those interested in digging through the archives and getting a first hand experience of Bourbaki's struggle with finding the “correct” axioms (as described in section 14. of Moore's paper), I recommend the Archives de l'Association des Collaborateurs de Nicolas Bourbaki. For a sample, see e.g. the Projet Cartan pour le début de la topologie where the equivalence of various axiomatisations is fleshed out.

  • 0
    @mathematrucker: thanks for this edit! I was unaware of this translation. I made it clearer that this edit and thus the translation wasn't made by me.2012-05-25
  • 0
    @mathematrucker: I have tried for the past 20 minutes to download the document. The "download" button seems inactive with MS Explorer. Using Google Chrome, it seems to work, but despite registering with a username and selecting a password (as requested) 4 consecutive times, I continue getting options to sign-up/register when I try to download it. Could you (or someone else) e-mail it to me, or perhaps post it in a more accessible location (such as in a sci.math post through [Math Forum](http://mathforum.org/kb/forum.jspa?forumID=13))?2012-05-25
  • 0
    I don't know if the problem is on my end, but I thought I should point out to anyone trying to download this paper that I just received the following message: *Threat Reason: Malware has been detected and reported.*2012-05-25
  • 0
    @mathematrucker: due to the malware reports by Dave L. Renfro I removed your addition to my answer. I hope you understand. Please check with the document host if everything is okay and you may want to check your own machine.2012-05-25
  • 0
    Pretty sure the above problems were server-related. Belated apologies. My translation of Kuratowski's Sur l'opération A¯ de l'Analysis Situs can now be found at https://www.academia.edu/13895470 plus many more references to the closure-complement theorem can be found at http://www.mathtransit.com2016-10-13
-4

Frechet is credited with the definition of a metric space in his 1906 paper and Hausdorff came up with the prototype of the standard axioms in his 1914 treatise on set theory.It was created as a direct abstraction of the metric space concept and it is not quite the modern definition. For example, the Hausdorff separation definition was one of the 4 axioms.

It becomes rather murky from that point who should be credited with the modern definition of a topological space.The basic concepts of both naive set theory and general topology seem to have begun to work their way into the mathematical discourse during the second decade of the 20th century in Europe, largely due to the oral lectures of Kuratowski and Alexandroff.

The modern definition of a function as a set of ordered pairs seems to have been popularized at the same time in this context (According to my extensive research on the history of the concept of function,it appeared first in print in a little known 1911 paper by Guiseppe Peano and was popularized by Kuratowski's aforementioned lectures.) The definition first appears in the monograph and textbook literature in the classical texts by these authors in 1920 and 1933 (?),respectively-as far as I know.

  • 1
    I've always seen metrics attributed to [Fréchet's thesis](http://dx.doi.org/10.1007/BF03018603) (published 1906). A linguistic question: "second decade of 20th century" = 1920ies? Could you please give references -- "a little known 1911 paper by Peano" is hard to locate.2011-10-08
  • 1
    @t.b. Thanks for the correction-it was indeed Maurice Frechet who defined the metric space.My bad and I edited it accordingly. The paper by Peano was referenced by Kuratowski in his famous book on set theory, "Set theory". It appears as a footnote after his definition of function as a set of ordered pairs which is single-valued.This reference by the Polish master,to the best of my knowledge as a non-expert, is the earliest published reference to this description of a function.2011-10-08
  • 1
    Seems my favorite anonymous fan has down-voted me again. Whoever you are,wish you'd have the courage to show yourself.2011-10-08
  • 5
    I downvoted because everything in this answer is either wrong (eg your misattribution of the definition of a metric space to Hausdorff), irrelevant to the question (eg the origin of the definition of a function as an ordered set), undocumented and speculative (your discussion of the oral lectures of Kuratowski and Alexandroff -- given your track record of false statements, I don't believe this without evidence) or contained in the very good answer above by t.b.2011-10-08
  • 1
    Who asked you? And frankly,your abusive and derogatory tone is becoming very irritating. My facts are correct except for the Hausdorff reference,which was careless and I gladly revised when it was pointed out to me.The Kuratowski book reference is accurate and precise. Whatever you need clarification on,kindly let me know-I didn't know I was being graded in what I assumed was an informal forum.2011-10-08
  • 10
    You asked. The Kuratowski book reference was irrelevant -- the question asked about the origin of the definition of a topological space, not the set-theoretic encoding of a function. And you didn't provide any real evidence to back up your claims. Finally, you are definitely being graded here, whether you want to be or not. Hence the voting system (and I'm certainly not the only person who has downvoted you). I'm particularly allergic to people attempting to pass themselves off as experts on subjects they know little about (for instance, your strong opinions on graduate level textbooks).2011-10-08
  • 0
    @Adam I discovered those facts largely through my research on the origins of the set-theoretic definition of a function,so in the context,it seemed relevant. I began with the footnotes in Kuratowski's text and expanded outward via the internet-a very important source being the 15 references of the excellent short biography of Kuratowski here: http://www.gap-system.org/~history/Biographies/Kuratowski.html. I would recommend most strongly the excellent article, L C Arboleda, Biography in Dictionary of Scientific Biography (New York 1970-1990).2011-10-08
  • 0
    @Adam(cont): However,all of these sources attribute the founding of the function concept to Kuratowski-I was a bit surprised to find the author himself in his set theory book attributes the definition to Peano.The rest on the history of topology comes from Boyer's classic A HISTORY OF MATHEMATICS and Morris Kline's outstanding MATHEMATICAL THOUGHT FROM ANCIENT TO MODERN TIMES-which I consider the definitive general text on the subject. But then I'm incompetent-what do I know?2011-10-08
  • 8
    I completely agree with @Adam that you should always back up **all** your claims with references automatically, not only after being asked for them. btw. Kuratowski and Mostowski [attribute the function concept to Peano](http://books.google.com/books?id=0BpMtVpkb7EC&pg=PA69), but but **not** to *Sulla definizione di funzione*, Atta Real An. Lin. 20 (1911) 3-5, but rather to *Formulaire des Mathématiques* (Torino, 1895). I believe that it's in §55 [on this page](http://www.archive.org/stream/formulairedemat01peangoog#page/n150/). But I also agree that this has little to do with the question.2011-10-08
  • 0
    @t.b. I didn't have the book in front of me and was working from memory.As I said,I didn't think I was getting graded in an informal forum. My point was most of what I know about the early history of topology I learned while researching this other,related question. And Adam has issues with me-that's become irritatingly clear.2011-10-08
  • 0
    @Adam Keep having fun down-voting me,I'm not getting into this childish game with you.2011-10-09
  • 4
    @Mathemagician1234 : My evil powers do not include the ability to vote multiple times on a question. The second -1 came from someone else. I should say that my voting philosophy is very simple -- I upvote things that are correct and helpful, and I downvote things that are wrong or misleading or irrelevant. If you would restrict your comments and answers to things that you know something about (and stop posting things that are about yourself rather than about math), then I would never have reason to downvote them.2011-10-09
  • 0
    @Adam I apologize for blaming you for the last one. And I DID post something about math-the question regarded the history of mathematics and I gave what I believed was an appropriate response.The single careless error was corrected by me when it was spotted. You don't agree,that's your right.2011-10-09
  • 5
    @Mathemagician1234 I sympathize with your situation. I think you are a person with good intentions who simply wishes to share his passion for mathematics (and, in this case, mathematics textbooks) on a public forum. Unfortunately, as I think I explained to you in another comment of mine recently, the world is not so simple. You see people (upvote and) downvote here based on the content of the answers. If someone downvotes you, then it does not mean that he "hates you" or that he wants you to go away from this forum or anything like that. A downvote simply reflects an opinion about the post ...2011-10-11
  • 1
    @Mathemagician1234 ... A downvote is not, and never should be, personal. I highly doubt, without knowing him, that Adam is downvoting your posts for "fun". The reasons for his downvote are very clearly explained in his comments. I understand that you are a passionate person and we unfortunately have too few such people in this world. However, I think you need to learn to comment and post in such a way that you are not constructing others here as "enemies". For example, you could write: "Could the downvoter please explain his reasons for downvoting?" ...2011-10-11
  • 1
    @Mathemagician1234 ... I think this would create an entirely different impression of yourself, a warmer expression, which people will appreciate. For example, when other posts are downvoted here, you might notice how the posters respond; they usually politely request the downvoter to explain the reasons for downvoting. Human nature means that it is hard *not to* feel guilty downvoting someone who has graciously accepted the downvote and politely requests the reasons for it. Of course, I am not suggesting that the downvotes that you have received on this website are all warranted ...2011-10-11
  • 1
    @Mathemagician1234 ... I think many people appreciate your efforts in posting here; I know I do. However, as I said, downvotes solely reflect an opinion about the content of the post and this has nothing to do with the poster's passion or efforts in posting. I also second t.b.'s advice regarding referencing your assertions. I know it is all too easy to remember a nice observation you had about the literature and to be too busy to check whether or not your reference is completely accurate. However, it is important to be precise, especially on a mathematics forum ...2011-10-11
  • 2
    @Mathemagician1234 ... I think this is something that one learns with time. Yes, to a certain extent this is an informal forum but "informal" does not mean "incorrect or inaccurate". "Informal" means that you can express your opinions about certain topics and are free to express ideas in your own (mathematically precise) manner. In any case, you may feel differently about this matter and you are, of course, free to post in your own manner. However, if you do not wish to receive downvotes, then I feel that it is worth considering the reasons others have given you for downvoting ...2011-10-11
  • 2
    @Mathemagician1234 ... I say this not because I think these reasons are "right or wrong" but because there is no point in asking for reasons if you do not wish to reflect on them. Finally, I wish to note that I have upvoted your answer; since there is only one other answer to this question with 16 more upvotes than this one, there will be no non-trivial consequences of a single upvote. The reason I am upvoting is because I appreciate your efforts and passion in posting here and I think that the absolute value of a negative vote count is no more relevant than the negativity of the vote count.2011-10-11
  • 0
    @Amitesh I thank you for your well intentioned words and advice.I mean that sincerely because you really didn't have to expend all that effort writing it. You have my gratitude for your good intentions and efforts. I should be more careful-but I come on this site to relax and sometimes,I don't do my due diligence. But "Adam" and others downvoted me 18 POINTS TODAY ALONE FOR NO APPARENT REASON. So you'll understand if I begin to get irritated. This is getting so childishly personal-and for what?!?2011-10-20
  • 0
    @Mathemagician1234 I notice that you have indeed received a few downvotes recently. I am sorry to hear that this is happening. If you feel that you are being incorrectly downvoted multiple times, then it could be a good idea to contact the moderators. However, as I am not sure of the situation, you would have to use your own judgement should you do this. I think part of the problem could be that some users see you as an "easy target" because they know that some others might support them in their downvoting of you and hence they are not afraid to continue downvoting you ...2011-10-20
  • 0
    @Mathemagician1234 ... If this is the case, then the situation is indeed becoming personal. However, this comment is now becoming off-topic; it is worth remembering that downvotes are, in the grand scheme of things, insignificant. My advice to you would be to simply ignore these downvotes; if people are indeed downvoting you for fun, then they are obviously looking for a reaction from you to their downvotes and you would be better off not giving them the satisfaction of seeing your reaction. I hope this helps.2011-10-20