Classifying homogeneous continua

Certain Homogeneous Unicoherent Indecomposable Continua

A simple closed curve is the simplest example of a compact, nondegenerate, homogeneous continuum. If a bounded, nondegenerate, homogeneous plane continuum has any local connectedness property, even of the weakest sort, it is known to be a simple closed curve [l, 2, 3].1 The recent discovery of a bounded, nondegenerate, homogenous plane continuum which does not separate the plane [4, 5] has give...

SOME THEOREMS ON n-HOMOGENEOUS CONTINUA

A Complete Classification of Homogeneous Plane Continua

We show that the only compact and connected subsets (i.e. continua) X of the plane R2 which contain more than one point and are homogeneous, in the sense that the group of homeomorphisms of X acts transitively on X, are, up to homeomorphism, the circle S1, the pseudo-arc, and the circle of pseudo-arcs. These latter two spaces are fractal-like objects which do not contain any arcs. It follows th...

Use of a New Technique in Homogeneous Continua

Recently Ungar has employed a theorem due to Effros [3] in the study of homogeneous continua. It is the purpose of this paper to use the same theorem to clarify and simplify Bing's proof that the only plane homogeneous continuum containing an arc is a simple closed curve [ 1 ]. Preliminary definitions and lemmas. Let M denote a continuum (= compact, connected, metric space). Then M is said to b...

More Monotone Open Homogeneous Locally Connected Plane Continua

This paper constructs a continuous decomposition of the Sierpiński curve into acyclic continua one of which is an arc. This decomposition is then used to construct another continuous decomposition of the Sierpiński curve. The resulting decomposition space is homeomorphic to the continuum obtained from taking the Sierpiński curve and identifying two points on the boundary of one of its complemen...