A complete classification of homogeneous plane continua

Abstract

We show that the only compact and connected subsets (i.e. continua) X of the plane $${\mathbb{R}^2}$$ 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 $${\mathbb{S}^1}$$ , 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 that any compact and homogeneous space in the plane has the form X × Z, where X is either a point or one of the three homogeneous continua above, and Z is either a finite set or the Cantor set. The main technical result in this paper is a new characterization of the pseudo-arc. Following Lelek, we say that a continuum X has span zero provided for every continuum C and every pair of maps $${f,g\colon C \to X}$$ such that $${f(C) \subset g(C)}$$ there exists $${c_0 \in C}$$ so that f(c 0) = g(c 0). We show that a continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable (i.e., every subcontinuum is indecomposable) and has span zero.