  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 $[0,1]$).
