Mapping complicated continua onto simple ones with small point inverses
by
Piotr Minc
Auburn University

A continuum X is decomposable if it is the union of two of its proper subcontinua, otherwise X is indecomposable. X is hereditarily indecomposable if it does not contain a decomposable subcontinuum.

We will talk about mapping indecomposable continua to dendroids (tree-like arcwise connected continua). Dendroids are supposed to be relatively simple. After all, they do not contain indecomposable subcontinua. Yet, we can prove that every chainable continuum (including the pseudoarc, which is hereditarily indecomposable) can be mapped into a dendroid such that all point-inverses consist of at most three points.

We will also talk about bottlenecks in dendroids. A subcontinuum of a dendroid D is a bottleneck if it intersects every arc connecting two nonempty open subsets of D. It turns out that every dendroid has a point p contained in arbitrarily small bottlenecks. Moreover, every plane dendroid contains a single point bottleneck. This implies, for instance, that each map from an indecomposable continuum into a plane dendroid must have an uncountable point-inverse.

Date received: February 26, 2003

Copyright © 2003 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 # cajz-69.