Disorder and topological robustness
Topological properties can survive many imperfections because small perturbations cannot change a discrete invariant continuously.
$$\text{invariant changes}\Rightarrow\text{gap or mobility gap closes}$$
What this means
Topological properties can survive many imperfections because small perturbations cannot change a discrete invariant continuously.
Deep dive: mathematical and physical mechanism
A sufficiently small perturbation preserving a protecting symmetry and the relevant spectral or mobility gap cannot change an integer or $\mathbb Z_2$ index. In disordered systems, real-space formulations and noncommutative Chern numbers can replace momentum-space formulas.
How it appears in quantum phenomena
Quantum Hall plateaus remain robust across disorder ranges.
Assumptions, caveats and limitations
Topology does not make devices immune to arbitrary heating, dephasing, interactions, or strong disorder.
Connect the ideas
For context, see the Topological Band Theory topic guide and the reading references.