Meager composants of tree-like continua

TL;DR

This paper investigates the structure of meager composants in tree-like continua, proving they are either one or continuum in number, and establishing a decomposition theorem relating to their topological properties.

Contribution

It demonstrates that no proper open meager composant exists in tree-like continua and characterizes the space of meager composants as a dendrite under certain conditions.

Findings

No proper open meager composant in tree-like continua

Number of meager composants is either 1 or continuum

Meager composants form a dendrite under specified conditions

Abstract

A subset $M$ of a continuum $X$ is called a \textit{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 Ordo\~{n}ez, we show that no tree-like continuum has a proper open meager composant, and that every tree-like continuum has either $1$ or $2^{\aleph_0}$ 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.