Accessibility of countable sets in plane embeddings of arc‐like continua

Abstract We consider the problem of finding embeddings of arc‐like continua in the plane for which each point in a given subset is accessible. We establish that, under certain conditions on an inverse system of arcs, there exists a plane embedding of the inverse limit for which each point of a given countable set is accessible. As an application, we show that for any Knaster continuum , and any countable collection of composants of , there exists a plane embedding of in which every point in the union of the composants in is accessible. We also exhibit new embeddings of the Knaster bucket‐handle continuum in the plane which are attractors of plane homeomorphisms, and for which the restriction of the plane homeomorphism to the attractor is conjugate to a power of the standard shift map on .