In view of Zhen Lin’s comment, I’ve written up both of our comments as an answer, so that we can get this question off the unanswered list.
The idea is that the domain is $X\cup\wp(X)\cup\wp(\wp(X))$; members of $X$ are the points of the space, members of $\wp(X)$ are sets of points in the space, and members of $\wp(\wp(X))$ are sets of sets of points in the space. One of the axioms will say that there is a member of $\wp(\wp(X))$ that is a topology. The relations $R$ and $S$ are there to make it possible to say that a type $0$ object (a point) is an element of a type $1$ object (a set of points), and that a type $1$ object is a member of a type $2$ object (a set of sets of points).
Note, though, that as Zhen Lin pointed out in the comments, the class of topological spaces is not an elementary class: no matter how you formalize it in first-order terms, your axioms will have models that aren’t topological spaces.