Abstract Given a metric continuum X, we say that X is 1 2 -homogeneous provided that X has exactly two orbits under the action of the group of homeomorphisms of X onto itself. In this paper, we introduce the hyperspace C 1 2 ( X ) consisting of all 1 2 -homogeneous subcontinua of X. We study compactness and connectedness of C 1 2 ( X ) for some classes of continua, mainly on finite graphs and dendroids, and we characterize the arc and the simple closed curve by using the structure of this hyperspace. Also, we show that every decomposable irreducible 1 2 -homogeneous continuum has one closed orbit.

中文翻译：

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闭合轨道和 12 齐次连续体的超空间

摘要 给定一个度量连续统 X，我们说 X 是 1 2 -齐次的，前提是 X 在 X 的同胚群作用下正好有两个轨道。在本文中，我们介绍了由 X 的所有 1 2 -齐次子连续体组成的超空间 C 1 2 ( X )。我们研究了 C 1 2 ( X ) 对于某些连续体类别的紧致性和连通性，主要是在有限图和树状体上，我们利用这个超空间的结构来刻画圆弧和简单的闭合曲线。此外，我们表明每个可分解的不可约 1 2 -齐次连续体都有一个闭合轨道。