A homogeneous continuum without the property of Kelley
by
Wlodzimierz J. Charatonik
University of Wroclaw and Universidad Nacional Autónoma de México

A Hausdorff continuum X is said to have the property of Kelley if for any subcontinuum K of X, any point p in K, and any open neighborhood U of K in C(X) (the hyperspace of all nonempty subcontinua of X) there is a neighborhood V of p in X such that for any q in V there is a continuum L satisfying q in L in U.

This property was defined for metric continua by J. L. Kelley and named Property (3.2) in J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), p. 22-36. Kelley has shown that continua with this property have contractible hyperspaces. Then basic properties of the property of Kelley were studied in R. W. Wardle, On a property of J. L. Kelley, Houston J. Math. 3 (1977), p. 291-299. In particular, he has shown that homogeneous continua have the property of Kelley. Here we construct an example showing that this is not true for Hausdorff continua.

We use an example by S. B. Nadler, Jr. of a continuum Y such that Y has, while Y ×Y does not have, the property of Kelley. A homogeneous Hausdorff continuum X is constructed such that it can be mapped onto Y by a monotone map. Since monotone (even confluent) maps preserve the property of Kelley, we conclude that X ×X is a homogeneous continuum that can be mapped onto Y ×Y by a monotone map, and thus X ×X does not have the property of Kelley.

Date received: January 31, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaa-23.