Universality of quantum Hall effect: Topological invariant and observable.
We show a universality in the quantum Hall effect in that the flux-averaged Hall conductivity in any two- or three-dimensional system with or without disorder is given by a topological invariant quantized in units of $\frac{{e}^{2}}{h}$ at $T=0$ for every energy level in a finite system. Relevance to the observable Hall conductivity over the whole energy spectrum is presented together with numerical results for lattice systems.