On a characterization of 𝑊-sets and the dimension of hyperspaces

Abstract

A subcontinuum A A of a continuum X X is a W W -set if for each mapping f : Y ↠ X f:Y \twoheadrightarrow X of an arbitrary continuum Y Y onto X X there is a continuum in Y Y which is mapped by f f onto A A . We characterize W W -sets in terms of accessibility by small continua. We localize several known results on continua all of whose subcontinua are W W -sets. Finally, we extend a result of J. T. Rogers by proving that if X X is an atriodic continuum whose first Čech cohomology group is finitely generated then the hyperspace C ( X ) C(X) of subcontinua of X X is two dimensional.