A Characterization of Chainable Continua

Abstract

In this paper, certain results of Bing (1) and myself (2) are extended. It is well-known that a chainable compact metric continuum must be a-triodic (contain no triods), hereditarily unicoherent (the common part of each two subcontinua is connected), and each subcontinuum must be chainable. Our principal result states that a compact metric continuum M is chainable if and only if M is a-triodic, hereditarily unicoherent and each indecomposable subcontinuum of M is chainable. Some condition on the indecomposable subcontinua of M seems essential, if we consider the dyadic solenoid, 5, which is indecomposable, a-triodic and hereditarily unicoherent. Indeed, each proper subcontinuum of S is an arc. However, S is not chainable, since it cannot be embedded in the plane.