  A universal topological marker has been proposed recently to map the
topological invariants of Dirac models in any dimension and symmetry class to
lattice sites. Using this topological marker, we examine the conditions under
which the global topological order, represented by the spatially averaged
topological marker, remains unchanged in the presence of disorder for 1D and 2D
systems. We find that if an impurity corresponds to varying a nonzero matrix
element of the lattice Hamiltonian, regardless the element represents hopping,
chemical potential, pairing, etc, then the average topological marker is
conserved. However, if there are many strong impurities and the average
distance between them is shorter than a correlation length, then the average
marker is no longer conserved. In addition, strong and dense impurities can be
used to continuously interpolate between one topological phase and another. A
number of prototype lattice models including Su-Schrieffer-Heeger model, Kitaev
chain, Chern insulators, Bernevig-Hughes-Zhang model, and chiral p-wave
superconductors are used to elaborate the ubiquity of these statements.
