Reduced maps and k-to-1 cut sets
by
Van Nall
University of Richmond

A map between continua is reduced if no proper subcontinuum of the range has a connected preimage. If a continuum is the image of a k-to-1 map from a continuum, then it must contain the image of a reduced k-to-1 map since a reduced k-to-1 map is one which is not k-to-1 on any proper subcontinuum of its domain. A k-to-1 cut set A of a continuum X is a finite subset of X such that X \A has at least k|A| components. We show that the image of a reduced at most k-to-1 map can not contain a k-to-1 cut set, and that a graph is the image of a reduced k-to-1 map if and only if it does not contain a k-to-1 cut set or an endpoint.

Date received: January 22, 1998