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How do model theorists treat topological spaces as structures, i.e., what are the options for the domain, relations, and operations?

I've never done any topology, so I have only the definition to go on. I'm not even sure if you would want to take as the domain the set X of points of the space, the power set of X, only the open sets, or something else.

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    While not an answer it might still be interesting to note that can also do much topology by passing to the open-set lattice "by itself" (cf. stone duality). Basically this means studying complete Hayting algebras which, while still not given by a first order theory, do look much more algebraic. Another option, used in constructive mathematics, is to study the notion of a "covering relations" (cf. formal topology), which might also give an idea what a theory of topological spaces might look like.2012-07-19

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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.