A Generalization of Weak Confluence
by
Pamela D. Roberson
Stephen F. Austin State University

A property of mappings of continua is defined which is a generalization of the notion of weak confluence.

Definition: Let f be a continuous function from a continuum X onto a continuum Y and let K₁, K₂, K₃, ... , K_n be a collection of subcontinua of Y. Then f is weakly confluent with respect to the collection K₁, K₂, K₃, ... , K_n if and only if there exists a subcontinuum H of X such that f(H) is one of the continua K₁, K₂, K₃, ... , K_n. For each positive integer n, f is n-weakly confluent if and only if f is weakly confluent with respect to every collection consisting of n subcontinua of Y.

Theorem: If f is a mapping of a continuum X onto a graph G, then there exists a positive integer n such that f is n-weakly confluent. Furthermore, the smallest such n is J(G)+\beta(G)+1 where J(G) is the number of junction points of G and \beta(G) is the first Betti number of G.

Theorem: If X is a continuum which is an inverse limit of an inverse limit sequence {X_i, f_i} where for each i, X_i has the property that every mapping of a continuum onto X_i is n-weakly confluent, then X has the property that every mapping of a continuum onto X is n-weakly confluent.

Date received: March 1, 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-39.