Characterizing chainable, tree-like, and circle-like continua

Abstract

Abstract. 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]).

A combinatorial invariant for escape time Sierpiński rational maps

“…We claim that C ∞ is a circle-like continuum. To see this, we use the following characterization given in [2]. The set C ∞ is known as the extended ray of angle 0.…”