The Osgood-Schoenflies Theorem Revisited
by Laurent Siebenmann
Math'ematique, B^at. 425, Universit'e de Paris-Sud, 91405-Orsay, France
http://topo.math.u-psud.fr/~lcs/contact
This retrospective article presents an elementary, and hopefully direct
and clear, geo- metric proof of what is usually called the (classical
planar) Schoenflies Theorem; it is stated as (ST) in x4 below _ with
mention of its early history, including W.F. Osgood's rarely cited
contributions. This (ST) is essentially the fact _ surprising in view
of known fractal curves _ that every compact subset of the cartesian
plane R2 that is homeomorphic to the circle S1, is necessarily the
frontier in R2 of a set homeomorphic to the 2-disk. Beware that the
`Generalized Schoenflies theorem' of B. Mazur [Maz] and M. Brown
[Brow1] _ proved five decades later and valid in all dimensions _ does
not imply (ST) since it assumes a condition of flatness (or local
flatness [Brow2]).