A continuous circle of pseudo-arcs filling up the annulus

Abstract

Abstract. We prove an early announcement by Knaster on a decomposition of the plane. Then we establish an announcement by Anderson saying that the plane annulus admits a continuous decomposition into pseudo-arcs such that the quotient space is a simple closed curve. This provides a new plane curve, "a selectible circle of pseudo-arcs", and answers some questions of Lewis.In 1922 the famous construction of an hereditarily indecomposable plane continuum was presented [6] by B. Knaster. Twenty-five years later Moise [11] constructed an hereditarily equivalent and hereditarily indecomposable plane continuum, and called it a pseudo-arc. Further, as a consequence of Bing's [3] characterization of the pseudo-arc, it turned out to be topologically equivalent to the Knaster curve. Moise's result was a starting point to intensive research on this very special continuum by a number of authors (see the survey paper [8]). In this paper we refer to some investigations made by Knaster before Moise's construction. Among the results obtained by Knaster during World War II one can find the following announcement, originally presented in Kiev in 1940:There exists a real-valued, monotone mapping from the plane that is not constant on any arc.In the construction Knaster's hereditarily indecomposable continua were exploited. Actually, Knaster's result can be reformulated in the following stronger version (see [10], p. 225):There exists a real valued, monotone mapping from the plane such that all pointinverses are hereditarily indecomposable.Unfortunately, Knaster's notes concerning this result were burned during the war, Knaster had never written down the result again, and even his closest exstudents do not know his original idea of construction.A result similar to that announced by Knaster (with higher dimensional analogues) was proved by Brown [5] in 1958. In fact, a continuous decomposition into hereditarily indecomposable continua of each Euclidean n-space with one point deleted was constructed, such that the real line was the quotient space.Having no confidence that we follow Knaster's idea, in this paper we construct an example of an open mapping as in the announcement. Then we use it to obtain a continuous decomposition of the plane band (annulus) into pseudo-arcs, such that

On the Conjecture of Wood and projective homogeneity

“…The pseudo-arc is a homogeneous 1-dimensional compact and connected space first constructed by Knaster [25]. It is an object of intense research both in topology [35], [30] and dynamical systems [23], [30], [31], [32]. It is hereditarily equivalent and hereditarily indecomposable.…”