Glasnik matematički, Vol. 41 No. 1, 2006.
Original scientific paper
Hyperspaces with exactly two orbits
Sam B. Nadler Jr.
Patricia Pellicer-Covarrubias
Abstract
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.
Keywords
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
Hrčak ID:
3305
URI
Publication date:
24.5.2006.
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