Symmetries of two-point sets

A two-point set is a subset of the plane which meets every planar line in exactly two-points. We discuss the problem “What are the topological symmetries of a two-point set?”. Our main results assert the existence of two-point sets which are rigid and the existence of two-point sets which are invariant under the action of certain autohomeomorphism groups. We pay particular attention to the isometry group of a two-point set, and show that such groups consist only of rotations and that they may be chosen to be any subgroup of S1 having size less than c. We also construct a subgroup of S1 having size c that is contained in the isometry group of a two-point set.