An atriodic tree-like continuum with positive span
W. Ingram (1972)
Fundamenta Mathematicae
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W. Ingram (1972)
Fundamenta Mathematicae
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H. Cook (1974)
Fundamenta Mathematicae
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Taras Banakh, Zdzisław Kosztołowicz, Sławomir Turek (2011)
Colloquium Mathematicae
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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]).
H. Cook (1970)
Fundamenta Mathematicae
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H. Cook (1970)
Fundamenta Mathematicae
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James Davis, W. Ingram (1988)
Fundamenta Mathematicae
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Hisao Kato (1988)
Compositio Mathematica
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Lee Mohler, Lex Oversteegen (1984)
Fundamenta Mathematicae
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Lex Oversteegen, E. Tymchatyn (1983)
Fundamenta Mathematicae
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Dušan Repovš, Arkadij Skopenkov, Evgenij Ščepin (1996)
Colloquium Mathematicae
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We prove that if the Euclidean plane 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 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.
J. Krasinkiewicz (1974)
Fundamenta Mathematicae
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