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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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    The class of topological spaces is not an elementary class: see [here](http://math.stackexchange.com/questions/46656/why-is-topology-nonfirstorderizable). So if you want a theory that admits all topological spaces as models and _only_ topological spaces, you must look for a theory in something other than first-order logic.2012-01-10
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    Thank you. I was just curious what a structure would look like. I'm not sure if a topology is necessarily a set, but is the idea to let the domain be the union of X, P(X), and P(P(X)), so the topologies will just be individuals? I don't understand the point of the R and S binary relations; are these just for the different sorts2012-01-10
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    @Rachel, in order to make sure a user reads a comment you leave, you can use the "@user" construction, so they will get a notification. It's not clear Zhen Lin will ever reread this without a notification.2012-01-10
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    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).2012-01-11
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    @ZhenLin Thanks!2012-01-11
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    @Brian Right, thanks. I was just wondering if the same could not be accomplished in a one-sorted language with regular set membership. Also, I am not sure how to mark my question as answered if no one leaves a separate answer.2012-01-11
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    You can ask @Zhen Lin to convert his comment to an answer; if he’s too busy or would rather not, you can synthesize an answer from the comments and answer it yourself.2012-01-11
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    @Rachel I think Brian's comment is closest in spirit to the answer you're looking for; my comment was more a subtle suggestion that the question needed clarification.2012-01-11
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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.