Partitions of open and connected subsets of the sphere
by
Lex Oversteegen
UAB
Coauthors: R. Fokkink, J. Mayer and E. Tymchatyn

Suppose that U is a connected and open subset of the sphere. It follows from a result of Kulkarni and Pinkall that U can be partitioned into disjoint closed subsets F_a of U such that for each a there exists a unique, round ball B_a ⊂ U (in the spherical metric) such that F_a ⊂ B_a. Each component of the boundary of each F_a in U is part of a round circle and hence is a particularly simple crosscut of U. If the boundary of U is connected, then the collection of all such crosscuts is sufficient for the study of prime ends of U. In this case we can replace these crosscuts by hyperbolic geodesics (or, as was done by H. Bell, by straight line segments) obtaining other partitions of U into disjoint closed sets. These results can be used to extend certain homeomorphisms on the boundary of U over U. (For example, the Schoenflies Theorem follows immediately.)

Date received: February 28, 2008