Origins of the modern definition of topology

Question

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.

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 06 '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 06 '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 06 '11 at 22:59

score 31 · Accepted Answer · edited Feb 21 '25 at 07:03

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:

C. Kuratowski, Sur l'opération $\overline{A}$ de l'Analysis Situs, Fund. Math. 3 (1922), 182–199 contains the formulation of the closure axioms — and the famous closure complement theorem showing that the axioms are independent of each other.
H. Tietze, Beiträge zur allgemeinen Topologie. I Axiome für verschiedene Fassungen des Umgebungsbegriffs, Math. Ann. 88 (1923), 3-4, 290–312 contains a definition in terms of open sets and neighborhoods but is somewhat more complicated.
P. Alexandroff, Zur Begründung der $n$-dimensionalen mengentheoretischen Topologie, Math. Ann. 94 (1926), Number 1, 296-308 (open sets)
The axioms are simply 1. that the intersection of two open sets “Gebiete” and arbitrary unions of open sets are open. 2. Hausdorff's separation condition.

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.

@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
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.com — mathematrucker, Oct 13 '16 at 15:12

Origins of the modern definition of topology

1 Answers1

Linked