# Topological spaces as model-theoretic structures — definitions?

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
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

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).