Characterizing indecomposable plane continua from their complements

Abstract:

We show that a plane continuum $X$ is indecomposable iff $X$ has a sequence $(U_n)_{n=1}^\infty$ of not necessarily distinct complementary domains satisfying the double-pass condition: for any sequence $(A_n)_{n=1}^\infty$ of open arcs, with $A_n \subset U_n$ and $\overline {A_n}\setminus A_n \subset \partial U_n$, there is a sequence of shadows $(S_n)_{n=1}^\infty$, where each $S_n$ is a shadow of $A_n$, such that $\lim S_n=X$. Such an open arc divides $U_n$ into disjoint subdomains $V_{n,1}$ and $V_{n,2}$, and a shadow (of $A_n$) is one of the sets $\partial V_{n,i}\cap \partial U$.