Choosing a Master Thesis Topic in Topology

Question

its my first time here. I’m a master student of mathematics and I’m trying to find a good topic for my thesis. I like it to be in Topology, but I don’t know exactly how to find one.

Answer 1

There are a lot of open problems you could write about. Check out books like "Open Problems in Topology" and "Recent Progress in Topology" (there are several parts from different years).

I like the simple statement of the Toronto space problem, e.g. :

Is there an infinite space $X$ that is Hausdorff and such that all $ Y \subseteq X$ with $|Y|=|X|$ are homeomorphic to $X$, except a discrete space?

If such a non-discrete space exists it's called a "Toronto space", as this is where the problem "arose"/ was invented. For $X$ countable it's classic that only the discrete space works so no countable Toronto spaces exist, but even one of size $\aleph_1$ would be very impressive, I think. There are some known properties that any Toronto space must have if it exists, etc.

But that is just one of the many fascinating open problems that's still out there.

Or you could focus on a recent paper and look at the results and questions leading to that.