Characterizing chainable, tree-like, and circle-like continua
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]).