The misunderstanding many have is that topology just the study of topological spaces. It is really also about continuous functions between two topological spaces.
If one has an infinite lattice model, where the Hilbert spaces is more-or-less $\ell^2(\mathbb{Z}^d)$, and if the Hamiltonian is periodic (a huge limitation) then one has momentum space that is naturally a torus $\mathbb{Z}^d$. This is for non-interacting Fermions.
If the Hamiltonian is gapped at $E_0$ then you get another topological space. For simplicity, assume $H$ has spectrum only at $0$ and $1$. Then over in momentum space the Hamiltonian becomes a continuous function from the torus to the set of all $k$-by-$k$ matrices that are Hermitian and with spectrum only in $0$ and $1$. (I take $k$ finite, limiting this even more.) This is a Grassmann manifold.
Thus we naturally find our Hamiltonian leads to a continuous function from a torus to a Grassmann manifold. In fact, this will be differentiable, and so what is often performed is often calculations using geometry. Also, many prefer to talk about a vector bundle, which is an equivalent view point.
That is the historical picture. Many other topological spaces arise in the study of a topological insulator, and often what is needed is more that a normal topological space. Finally, many more modern calculations involve operator theory or operator algebras, so there is only noncommutative topology where there are not really any topological spaces at all. These sorts of things are needed to get beyond infinite, perfect models.