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Displaying similar documents to “An uncountable collection of mutually exclusive planar atriodic tree-like continua with positive span”

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

Taras Banakh, Zdzisław Kosztołowicz, Sławomir Turek (2011)

Colloquium Mathematicae

Similarity:

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]).

On uncountable collections of continua and their span

Dušan Repovš, Arkadij Skopenkov, Evgenij Ščepin (1996)

Colloquium Mathematicae

Similarity:

We prove that if the Euclidean plane ℝ 2 contains an uncountable collection of pairwise disjoint copies of a tree-like continuum X, then the symmetric span of X is zero, sX = 0. We also construct a modification of the Oversteegen-Tymchatyn example: for each ε > 0 there exists a tree X ⊂ ℝ 2 such that σX < ε but X cannot be covered by any 1-chain. These are partial solutions of some well-known problems in continua theory.