Almost isometric flat spaces and perturbation-equivalence

We construct two non-isometric closed subsets of the real line which are almost isometric, and show that any similar example in an Euclidean space is essentially one-dimensional. We then define perturbation-equivalence of almost isometric embeddings, and find a rigid closed subset of the line with an almost isometry onto itself which is not a perturbation of the identity. Finally we show that any almost isometry from an Euclidean space to itself is a perturbation of a sequence of isometries.