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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.

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According to Wikipedia it was Kuratowski in 1922. – t.b. Oct 6 '11 at 21:11
Hm. This is very strange. Kuratowski wrote down his closure axioms in Sur l'opération $\overline{A}$ de l'Analysis Situs 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 which is manifestly nonsense, as this is about finite sets. – t.b. Oct 6 '11 at 21:50
It seems to me that this paper 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, Historia Mathematica 35 (2008) 220–241. – t.b. Oct 6 '11 at 22:32
@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. – LostInMath Oct 6 '11 at 22:59
up vote 23 down vote accepted

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.

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@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. – t.b. May 25 '12 at 17:08
@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)? – Dave L. Renfro May 25 '12 at 17:34
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. – Dave L. Renfro May 25 '12 at 17:44
@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. – t.b. May 25 '12 at 19:16

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.

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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. – Adam Smith Oct 8 '11 at 4:07
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). – Adam Smith Oct 8 '11 at 4:34
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, 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. But I also agree that this has little to do with the question. – t.b. Oct 8 '11 at 9:33
@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. – Adam Smith Oct 9 '11 at 3:27
@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 ... – Amitesh Datta Oct 11 '11 at 10:54

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