Characterizing chainable, tree-like, and circle-like continua

TL;DR

This paper characterizes tree-like, circle-like, and chainable continua by their ability to be mapped onto simpler structures like trees, circles, or intervals via open cover-based maps, providing a new perspective on continuum classification.

Contribution

It establishes a characterization of certain continua through open cover-based maps onto fundamental topological structures, linking local cover properties to global shape.

Findings

Characterization of tree-like, circle-like, and chainable continua.

Equivalence between continuum types and existence of specific open cover maps.

Provides criteria for acyclic curves via open cover maps.

Abstract

We prove that a continuum $X$ is tree-like (resp. circle-like, chainable) if and only if for each open cover $\U_4=\{U_1,U_2,U_3,U_4\}$ of $X$ there is a $\U_4$-map $f:X\to Y$ onto a tree (resp. onto the circle, onto the interval). A continuum $X$ is an acyclic curve if and only if for each open cover $\U_3=\{U_1,U_2,U_3\}$ of $X$ there is a $\U_3$-map $f:X\to Y$ onto a tree (or the interval $[0,1]$).