TITLE: Model Theory and Topology Connections
QUESTION [11 upvotes]: 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.

REPLY [4 votes]: Most obvious examples come from considering the Stone space of complete types over a complete theory. For instance:

Compactness of the Stone space is closely related to compactness of first-order logic.
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).
Formulas, and definable sets correspond exactly to clopen subsets of the Stone space.
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.
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:

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).
For countable models, the group is, in fact, Polish.
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.