Characterizing indecomposable plane continua from their complements

TL;DR

This paper characterizes indecomposable plane continua by a novel sequence-based condition involving their complementary domains, providing a new criterion for indecomposability.

Contribution

It introduces the double-pass condition as a new characterization of indecomposable plane continua based on their complements.

Findings

The double-pass condition is equivalent to indecomposability.

A sequence of complementary domains satisfying this condition characterizes indecomposable continua.

Provides a new topological criterion for analyzing plane continua.

Abstract

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.