Characterizing indecomposable plane continua from their complements

Proceedings of the American Mathematical Society. Volume 136, Number 11, November 2008, Pages 4045--4055. We show that a plane continuum X is indecomposable iff X has a sequence (U_n) of not necessarily distinct complementary domains satisfying what we call the double-pass condition: If one draws an open arc A_n in each U_n whose ends limit into the boundary of U_n, one can choose components of U_n minus A_n whose boundaries intersected with the continuum (which we call shadows) converge to the continuum.