Retractible Continua
by
Alejandro Illanes
Universidad Nacional Autonoma de Mexico

A continuum X is said to be retractible provided that each of its subcontinua is a retract of X. Retractible continua were defined and studied by J. J. Charatonik in 1986. In his paper, he posed the following problem:

PROBLEM. Give a structural characterization of retractible continua.

This problem seems to be very difficult. However, adding requierements to the retractions, some characterizations can be obtained.

A continuum X is said to be d-retractible (resp. sd-retractible, m-retractible, c-retractible, o-retractible), provided that for each subcontinuum A of X, there exists a deformation ( resp. strong deformation, monotone, confluent, open) retraction from X onto A. In this talk, the following results will be presented:

Theorem. The following assertions are equivalent:

1. X is a dendrite,
2. X is d-retractible and,
3. X is sd-retractible.

Theorem. If X is a pathwise connected c-retractible continuum, then X is hereditarily locally connected.

Theorem. If X is a pathwise connected continuum, then the following assertions are equivalent:

1. X is o-retractible and,
2. X is homeomorphic to an interval or to a simple closed curve.

It will be also presented an example of a nonlocally connected retractible dendroid and some other examples related to this topic.

Date received: June 24, 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 Mathematical Conference Abstracts. Document # caah-42.