Tree-likeness of dendroids and λ-dendroids
H. Cook (1970)
Fundamenta Mathematicae
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H. Cook (1970)
Fundamenta Mathematicae
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Wojciech Dębski, J. Heath, J. Mioduszewski (1992)
Fundamenta Mathematicae
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It is known that no dendrite (Gottschalk 1947) and no hereditarily indecomposable tree-like continuum (J. Heath 1991) can be the image of a continuum under an exactly 2-to-1 (continuous) map. This paper enlarges the class of tree-like continua satisfying this property, namely to include those tree-like continua whose nondegenerate proper subcontinua are arcs. This includes all Knaster continua and Ingram continua. The conjecture that all tree-like continua have this property, stated...
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]).
Hisao Kato (1990)
Fundamenta Mathematicae
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W. Ingram (1974)
Fundamenta Mathematicae
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C. Eberhart, J. Fugate (1971)
Fundamenta Mathematicae
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Hisao Kato (1988)
Compositio Mathematica
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J. Fugate (1971)
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.
Lex Oversteegen, E. Tymchatyn (1983)
Fundamenta Mathematicae
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Lee Mohler, Lex Oversteegen (1984)
Fundamenta Mathematicae
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