lew (12/04/82)
Thanks to Spencer Thomas for posting Hausdorff's area paradox. I had heard of this and wondered about it. I think I understand the derivation in outline at least. I am puzzled, though, by Spencer's closing remark: "Take heart, though, area still works in 2-space." By "2-space" does he mean the infinite plane? If so, I would remark that the conventional definition of area still works on the sphere, too. Note that the sets used to form the paradox (A, B, and C) are very strange. Each of them covers the sphere. Every point of the sphere has infinitely many points of A, B, and C arbitrarily close to it. I think that if we confine the definition of sets to the open sets of conventional topology (Sets such that the neighborhood of every point in the set contains only points of the set,) area works out fine and conforms to conventional usage. Lew Mammel, Jr. ihuxr!lew