Local constancy of the Fredholm index

Near a Fredholm operator, split the domain into its kernel and orthogonal complement and the target into its range and orthogonal complement. The large invertible block stays invertible under small norm perturbations. Invertible block elimination reduces the perturbed operator to that block and a map between the finite-dimensional kernel and cokernel spaces. Its index is their dimension difference regardless of its rank. Thus the Fredholm set is open and its index is locally constant.