Origins of the modern definition of topology 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.

 A: 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.
