Singularities of meager composants and filament composants

TL;DR

This paper investigates the topological structure of certain continua, showing under specific conditions that a union of nowhere dense subcontinua forms a composant of an indecomposable continuum, with applications to chainable and homogeneous continua.

Contribution

It proves that under particular conditions, the union of all nowhere dense subcontinua containing a point is homeomorphic to a composant of an indecomposable continuum, extending previous examples.

Findings

X is homeomorphic to a composant of an indecomposable continuum.

If Y is chainable or an inverse limit of topological graphs, then Y is indecomposable and X is a composant.

Results relate to a 2007 question on homogeneous continua.

Abstract

Suppose $Y$ is a continuum, $x\in Y$, and $X$ is the union of all nowhere dense subcontinua of $Y$ containing $x$. Suppose further that there exists $y\in Y$ such that every connected subset of $X$ limiting to $y$ is dense in $X$. And, suppose $X$ is dense in $Y$. We prove $X$ is homeomorphic to a composant of an indecomposable continuum, even though $Y$ may be decomposable. An example establishing the latter was given by Christopher Mouron and Norberto Ordo\~nez in 2016. If $Y$ is chainable or, more generally, an inverse limit of identical topological graphs, then we show $Y$ is indecomposable and $X$ is a composant of $Y$. For homogeneous continua we explore similar problems which are related to a 2007 question of Janusz Prajs and Keith Whittington.