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.

Solution 1:

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

Solution 2:

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