Some topological extensions of plane geometry

Let A be a closed subset of the plane. For a point b in the plane whose distance to A is d(b,A) let S(b,A) = {z : |z-b| d (b,A)}. Let E(A) be the set of points e for which S(b,A) ∩ A has at least two points. Let f(A) be the set of open intervals (a,b) for which there is an e ℇ E(A) and a component of S(e,A)-A with endpoints a and b. The sets E(A) and f(A) are the central tools in this paper.