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

https://hrcak.srce.hr/3305

Publication date:

24.5.2006.

Visits: 1.648 *