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Hyperspaces with exactly two orbits

Croatian Mathematical Society and Department of Mathematics, University of Zagreb
Publication Date
  • Arc
  • Arc-Like
  • Atriodic
  • Chainable
  • Circle-Like
  • Connected Im Kleinen
  • Continuum
  • Finest Monotone Map
  • Hilbert Cube
  • Homogeneous
  • 1/2-Homogeneous
  • 1/N-Homogeneous
  • Hyperspace
  • Layers (Tranches)
  • Locally Connected
  • N-Cell
  • N-Fold Hyperspace
  • N-Fold Symmetr


Let C(X) be the hyperspace of all subcontinua of a (metric) continuum X. It is known that C(X) is homogeneous if and only if C(X) is the Hilbert cube. We are interested in knowing when C(X) is 1/2-homogeneous, meaning that there are exactly two orbits for the action of the group of homeomorphisms of C(X) onto C(X). It is shown that if X is a locally connected continuum or a nondegenerate atriodic continuum, and if C(X) is 1/2-homogeneous, then X is an arc or a simple closed curve. We do not know whether an arc and a simple closed curve are the only continua X for which C(X) is 1/2-homogeneous.

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