Glasnik Matematicki, Vol. 41, No.1 (2006), 141-157.

HYPERSPACES WITH EXACTLY TWO ORBITS

Sam B. Nadler, Jr. and Patricia Pellicer-Covarrubias

S .B. Nadler, Jr., Department of Mathematics, West Virginia University, P. O. Box 6310, Morgantown, WV 26506-6310, USA
e-mail: nadler@math.wvu.edu

P. Pellicer-Covarrubias, Department of Mathematics, West Virginia University, P. O. Box 6310, Morgantown, WV 26506-6310, USA
Departamento de Matemáticas, Facultad de Ciencias, Circuito Exterior, Ciudad Universitaria, México, D. F., C. P. 04510, México
e-mail: paty@ciencias.unam.mx
e-mail: paty@math.wvu.edu

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.

2000 Mathematics Subject Classification.   54B20, 54F15.

Key words and phrases.   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 symmetric product, orbit, property of Kelley, simple closed curve, simple triod, tree-like, triod.

Full text (PDF) (free access)

DOI: 10.3336/gm.41.1.12

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