A characterization of continua that contain no $n$-ods and no $W$-sets

Abstract:

A proper, nondegenerate subcontinuum $K$ of a continuum $Y$ is said to be a $W$-set if, for every continuum $X$ and map $f$ of $X$ onto $Y$, some subcontinuum of $X$ is mapped by $f$ onto $K$. Jim Davis asked whether a simple closed curve is the only atriodic continuum that contains no $W$-set. An affirmative answer is given to this question. The result follows as a corollary to the more general theorem that a continuum contains no $n$-od and has no $W$-set if and only if it is a graph in which every point is contained in a simple closed curve. Properties of this class of graphs are also described.