Characterizing chainable, tree-like, and circle-like continua
Volume 124 / 2011
Colloquium Mathematicum 124 (2011), 1-13
MSC: Primary 54F15, 54F50; Secondary 54D05.
DOI: 10.4064/cm124-1-1
Abstract
We prove that a continuum $X$ is tree-like (resp. circle-like, chainable) if and only if for each open cover $\mathcal U_4=\{U_1,U_2,U_3,U_4\}$ of $X$ there is a $\mathcal U_4$-map $f\colon X\to Y$ 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 $\mathcal U_3=\{U_1,U_2,U_3\}$ of $X$ there is a $\mathcal U_3$-map $f\colon X\to Y$ onto a tree (or the interval $[0,1]$).
