Characterizing indecomposable plane continua from their complements

Author:

Curry Clinton,Mayer John,Tymchatyn E.

Abstract

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

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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