We show that the only compact and connected subsets (i.e. continua) X of the plane $${\mathbb{R}^2}$$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 $${\mathbb{S}^1}$$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 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 $${f,g\colon C \to X}$$f,g:C→X such that $${f(C) \subset g(C)}$$f(C)⊂g(C) there exists $${c_0 \in C}$$c0∈C so that f(c0) = g(c0). 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.

中文翻译：

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齐次平面连续体的完整分类

我们证明了平面 $${\mathbb{R}^2}$$R2 的唯一紧致和连通子集（即连续体）X 包含一个以上的点并且是齐次的，因为X 在 X 上的传递作用是，直到同胚，圆 $${\mathbb{S}^1}$$S1、伪弧和伪弧的圆。后两个空间是不包含任何弧的分形对象。由此可知，平面中的任何紧齐齐空间都具有 X × Z 形式，其中 X 是一个点或上述三个齐次连续体中的一个，Z 是有限集或康托集。 主要技术结果本文是对伪弧的新表征。在 Lelek 之后，我们说连续统 X 具有为每个连续统 C 和每对映射 $${f,g\colon C \to X}$$f,g 提供的跨度零：C→X 使得 $${f(C) \subset g(C)}$$f(C)⊂g(C) 存在 $${c_0 \in C}$$c0∈C 使得 f(c0 ) = g(c0)。我们证明连续统与伪弧同胚当且仅当它是遗传不可分解的（即，每个子连续统都是不可分解的）并且跨度为零。