We prove that a continuum X is tree-like (resp. circle-like, chainable) if and only if for each open cover 𝓤₄ = {U₁,U₂,U₃,U₄} of X there is a 𝓤₄-map f: X → 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 𝓤₃ = {U₁,U₂,U₃} of X there is a 𝓤₃-map f: X → Y onto a tree (or the interval [0,1]).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-1-1,
author = {Taras Banakh and Zdzis\l aw Koszto\l owicz and S\l awomir Turek},
title = {Characterizing chainable, tree-like, and circle-like continua},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {1-13},
zbl = {1268.54011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-1-1}
}
Taras Banakh; Zdzisław Kosztołowicz; Sławomir Turek. Characterizing chainable, tree-like, and circle-like continua. Colloquium Mathematicae, Tome 122 (2011) pp. 1-13. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-1-1/