Help on a proof related to the boundary of a star-shaped set

["Yajun Wang"]

Hi there:

I have the following problem:

We have set A \subset R^2, and its complement B. If both of them are
star shaped, is it possible to show that their boundary(should be the
same) is homeomorphic to an open line segment?

There are other properties available, e.g, one of them is open set.
Does any one see similar results some where? I have been stuck with
this problem for weeks. Any help is highly appreciated.

Thanks.

regards,
yalding