λ Connectivity in the Plane

Abstract

Every arcwise connected nonseparating plane continuum has a hereditarily decomposable boundary. Hence, all arcwise connected nonseparating plane continua have the fixed point property. This chapter discusses that the generalization of arcwise connectivity provides another way of describing continua that satisfy the hypothesis of Bell's theorem. It is proposed that if M is a plane continuum that does not have infinitely many complementary domains. Then M is λ connected if every continuum in the boundary of M is decomposable. A plane continuum M is λ connected if each link in M is hereditarily decomposable. From this characterization, it follows that λ connected plane continua share several other properties with arcwise connected continua.