Continua Which Have The Property of Kelley Hereditarily
by
Gerardo Acosta-Garcia
Universidad Nacional Autonoma de Mexico
Coauthors: Alejandro Illanes

A continuum is a compact, connected, metric space. For a continuum X, C(X) will denote the set of all the subcontinua of X. A continuum X has the property of Kelley if for each a in X, each A in C(X) such that a in A and each sequence (a_n)_n in X converging to a, there exists a sequence (A_n)_n in C(X) converging to A such that a_n in A_n for every positive number n. In this paper we study the property of Kelley hereditarily. Namely, X has the property of Kelley hereditarily provided that each one of its subcontinua have the property of Kelley. Our main result is the following characterization of hereditarily locally connected continua.

A continuum X is hereditarily locally connected if and only if X is hereditarily of Kelley and arcwise connected.

This result generalizes the following theorem due to S. Czuba: a dendroid X is a dendrite if and only if, X is hereditarily of Kelley. Some other results are the followings:

If X has the property of Kelley but not the property of Kelley hereditarily, then X contains an \infty-od.

Each atriodic continuum with the property of Kelley, has the property of Kelley hereditarily.

If a metric compactification X of the space [0, 1) has the property of Kelley, then X has the property of Kelley hereditarily.

Confluent mappings preserves the property of Kelley hereditarily.

The hyperspaces C(X) and 2^X (of all non-empty closed subsets of X)does not have the property of Kelley hereditarily.

The product X×Y of continua X and Y does not have the property of Kelley hereditarily.

The property of Kelley hereditarily is a Whitney reversible property.

Date received: February 17, 1997

Copyright © 1997 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 # caam-06.