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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. 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. –  Zhen Lin Jan 10 '12 at 12:49
    
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 sorts –  Rachel Jan 10 '12 at 14:03
    
@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. –  Grumpy Parsnip Jan 10 '12 at 16:58
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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). –  Brian M. Scott Jan 11 '12 at 4:50
    
@ZhenLin Thanks! –  Rachel Jan 11 '12 at 5:11
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1 Answer

up vote 5 down vote accepted

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.

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