Meager composants of tree-like continua
David S. Lipham

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 of a continuum is called a \textit{meager composant} if is maximal with respect to the property that every two of its points are contained in a nowhere dense subcontinuum of . 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 or meager composants. We also prove a decomposition theorem: If is tree-like and every indecomposable subcontinuum of is nowhere dense, then the partition of into meager composants is upper semi-continuous and the space of meager composants is a dendrite.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms
