In the field of topological insulators
- What topological space do they talk of?
- Looking for some resources that sheds light on the topology part of topological insulator
Put "bluntly", in the IQHE for example, one refers to the space of classes of vector bundles over the two-dimensional torus (the Brillouin zone), which apparently is isomorphic to $\mathbb{Z}$, the isomorphism given by the Hall conductivity.
It really depends on what perspective you want, and how realistic you want to be from the physics side. Very generally you need to know algebraic topology, vector bundle theory, K-theory (of vector bundles or C-star algebras). There are plenty of resources about this already. I propose for you to start with the book of Prodan and Schulz-Baldes: https://www.springer.com/gp/book/9783319293509