[net.math] Hausdorff's area paradox

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