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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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    @t.b.: Indeed it gives the answer and much more. Quite interesting stuff. Thank you very much! According to the article, in 19$2$5 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

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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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    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