  We investigate the relation between the localization and the topological
properties of two-dimensional gapped quantum systems of independent electrons
in a disordered background, including magnetic fields, as in the case of Chern
insulators and Quantum Hall systems. We prove that the existence of a
well-localized generalized Wannier basis for the Fermi projection implies the
vanishing of the Chern marker, which is a natural generalization of the Chern
number to non-periodic insulators, and coincides with the Hall conductivity in
the linear response regime. Moreover, we state a localization dichotomy
conjecture for general non-periodic gapped quantum systems.
