# Model Theory and Topology Connections

I have studied a bit of model theory, when I say "a bit" I have studied much more than is available to a typical undergraduate in the UK (i think, certainly from what I have seen) but I am sure this is all introductory material when looked at on a bigger scale (as things always are) but it is the area that I am going to pursue in my masters degree next year and hopefully beyond.

I have also done some very introductory material in topology, sort of covered a typical "intro to topology" book.

What I was wanting to ask about was the connections between the two. Now I know that there are many strong connections, but I am yet to actually come across any of these connections (apart from reading things which says they exists and hearing people talk about these connections very generally).

So I was wondering if anyone could give me some idea as to where the connections lie and some references for reading about this. I am aware that there is something called "o-minimality" but I have no idea what this is, so I was looking for some good references. Also as I stated above I have an introductory grasp on topology so is it the case that I need to learn much more before I can grasp these connections at all?

(I would also like to add that I am not really sure what I am asking about, as I'm sure is obvious from the question, so please feel free to tell me that I'm looking for the wrong things or my questions is too naive.)

As always any help is greatly appreciated.

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What about first-order compactness being an instance of topological compactness? – Quinn Culver Dec 20 '12 at 21:49

You can start by reading Alan Dow's paper:

Dow, A. An introduction to applications of elementary submodels to topology. Topology Proc. 13 (1988), no. 1, 17–72. MR1031969

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Very good suggestion. The paper is freely available from Alan’s home page at UNCC. – Brian M. Scott Dec 20 '12 at 17:13
@Brian: Thanks for catching that mistake! – Asaf Karagila Dec 20 '12 at 17:36
You’re welcome. I used to know Alan, so it kind of hit me in the eye. – Brian M. Scott Dec 20 '12 at 17:49

Since you mention o-minimality I thought I should mention that this area of model theory which is closely related to (generalizations of?) semi-algebraic geometry and real-algebraic geometry. In particular, techniques from o-minimality have been used to make some progress towards solving the Andre-Oort Conjecture, I am referring to the paper by Jonathon Pila called "O-minimality and the Andre-Oort Conjecture for $\mathbb{C}^n$". For introductory papers on this subject, I suggest a paper by Thomas Scanlon called "O-minimality as an approach to the Andre-Oort conjecture" which can be found on Scanlon's academic webpage.

I would also like to mention another area of model theory with connection to analysis. It is called the Model Theory of Metric Structures. In this model theory, one changes the particular logic one is using and the definition of structure. Structures in this theory are now complete metric spaces, connectives are uniformly continuous real-valued functions, etc. You can find an introduction online (legally) for free. I forget the authors, but just search "Model Theory for Metric Structures."

Finally, there is also a thing called topological model theory, which was first proposed by Anand Pillay (I believe). There are lecture notes by Ziegler called "Topological Model Theory" published by Springer. I am sure you can find this in your university library.

Have fun

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Most obvious examples come from considering the Stone space of complete types over a complete theory. For instance:

1. Compactness of the Stone space is closely related to compactness of first-order logic.
2. Any subset of the Stone space can be omitted simultaneously if it is meager (and the proof I know of even the simplest version of omitting types theorem uses Baire category theorem).
3. Formulas, and definable sets correspond exactly to clopen subsets of the Stone space.
4. Morley rank of a formula equals the Cantor-Bendixson rank of the clopen set defined by it (within a Stone space over an $\aleph_0$-saturated structure), similarly the multiplicity.
5. Type-definable sets are closed subsets of the Stone space (though right now I'm not certain if the converse is true).

For a different, more geometrical flavour of examples:

1. The automorphism group of any model has natural structure of topological group (with open sets defined by fixing the images and preimages of finite tuples).
2. For countable models, the group is, in fact, Polish.
3. For saturated countable models it is isomorphic to the Baire group $\omega^\omega$.

For different yet kind of examples, as far as I know, the theory of algebraic groups generalizes to definable groups (of finite Morley rank, I think) in suitable theories (but I know next to nothing about it, so I will not elaborate).

A different yet kind of use of topology comes from some particular theories related to particular topological spaces. For example, if we want to show something about real closed fields, we often only need to work in the real numbers, and if we want to show something about algebraically closed fields of characteristic $0$, we only need to work with the complex numbers. Both of these come with their natural topology and it can sometimes be used to show some (first-order) facts which would otherwise be hard to prove.

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