Meager composants of tree-like continua

Abstract

A subset M of a continuum X is called a meager composant if M is maximal with respect to the property that every two of its points are contained in a nowhere dense subcontinuum of X. Motivated by questions of Bellamy, Mouron and Ordoñez, we show that no tree-like continuum has a proper open meager composant, and that every tree-like continuum has either 1 or 2ℵ0 meager composants. We also prove a decomposition theorem: If X is tree-like and every indecomposable subcontinuum of X is nowhere dense, then the partition of X into meager composants is upper semi-continuous and the space of meager composants is a dendrite.