Separation of plane sets by equidistant simple closed curves

We prove that if two subsets A and B of the plane are connected, A is bounded, and the Euclidean distance ρ(A,B) between A and B is greater than zero, then for every positive ε<ρ(A,B), the sets A and B can be separated by a simple closed curve (also known as a Jordan curve) whose points all lie at distance ε from the set A. We also prove that the ε-boundary of a connected bounded subset A of the plane contains a simple closed curve bounding the domain containing the open ε-neighbourhood of A. It is shown that in both statements the connectivity condition can be significantly weakened. We also show that the ε-boundary of a nonempty bounded subset of the plane contains a simple closed curve. This result complements Morton Brown's statement that the ε-boundary of a nonempty compact subset of the plane is contained in the union of a finite number of simple closed curves.