The label "topological material" has been fairly liberally applied to a wide class of materials, and there is not always a clear agreement what should qualify as topological. However, there are some features which are common across most topological materials. Such features include:
- Unusual surface states
- Quantized response functions
- Bulk quantized invariants
Though of course this is not an exhaustive list, not all types of topology have all of these features and these features are interdependent.
The quantum hall effect (in its many forms) are related to the second point on that list: Quantized response functions.
In particular, the famous TKNN invariant relates the hall conductivity $\sigma_{xy}$ (a response function) directly to a quantized topological invariant (the Chern number) through:
$$
\sigma_{xy} = -\frac{e^2}{2\pi \hbar}\sum_{\alpha}C_{\alpha}
$$
Where the sum is over the filled bands and $C_{\alpha}$ is the Chern number of band $\alpha$.
The topological invariant $C_{\alpha}$ is quantized to be an integer, and therefore the response function is quantized (in units of $e^2/{2\pi \hbar}$).
The interplay of response functions and topology is profound, as usually nontrivial topology is detected through some response-type experiment. There are many other examples, including the quantized circular photogalvanic effects (an optical response) associated with certain Weyl semimetals or the quantized magnetoelectric responses, associted with certain magnetic TIs
A good review of the interplay of response functions and topology can (in my opinion) be found in D. Vanderbilt's Berry Phase in electronic structure theory (2018), though most texts on topological insulators should cover this link.
