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 ▫$f: X ...\to Y$▫ 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 ▫$f: X \to Y$▫ onto a tree (or the interval ▫$[0,1]$▫).
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