Two-to-one maps on solenoids and Knaster continua
Wojciech Dębski (1992)
Fundamenta Mathematicae
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It is shown that 2-to-1 maps cannot be defined on certain solenoids, in particular on the dyadic solenoid, and on Knaster continua.
Wojciech Dębski (1992)
Fundamenta Mathematicae
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It is shown that 2-to-1 maps cannot be defined on certain solenoids, in particular on the dyadic solenoid, and on Knaster continua.
Sam Nadler (1972)
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...
Sam Nadler (1980)
Fundamenta Mathematicae
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Sam Nadler, J. Quinn (1973)
Fundamenta Mathematicae
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B. J. Pearson (1974)
Colloquium Mathematicae
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J. Krasinkiewicz, Piotr Minc (1979)
Fundamenta Mathematicae
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Eiichi Matsuhashi (2012)
Bulletin of the Polish Academy of Sciences. Mathematics
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We prove the following results. (i) Let X be a continuum such that X contains a dense arc component and let D be a dendrite with a closed set of branch points. If f:X → D is a Whitney preserving map, then f is a homeomorphism. (ii) For each dendrite D' with a dense set of branch points there exist a continuum X' containing a dense arc component and a Whitney preserving map f':X' → D' such that f' is not a homeomorphism.
Lex Oversteegen, E. Tymchatyn (1984)
Fundamenta Mathematicae
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Jo Heath, Van C. Nall (2006)
Fundamenta Mathematicae
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A bottleneck in a dendroid is a continuum that intersects every arc connecting two non-empty open sets. Piotr Minc proved that every dendroid contains a point, which we call a center, contained in arbitrarily small bottlenecks. We study the effect that the set of centers in a dendroid has on its structure. We find that the set of centers is arc connected, that a dendroid with only one center has uncountably many arc components in the complement of the center, and that, in this case,...
Carl Eberhart, Sam Nadler, William Nowell (1981)
Fundamenta Mathematicae
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Jo Heath, Van C. Nall (2003)
Fundamenta Mathematicae
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In 1940, O. G. Harrold showed that no arc can be the exactly 2-to-1 continuous image of a metric continuum, and in 1947 W. H. Gottschalk showed that no dendrite is a 2-to-1 image. In 2003 we show that no arc-connected treelike continuum is the 2-to-1 image of a continuum.
Hanna Patkowska (1963)
Fundamenta Mathematicae
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B. J. Pearson (1975)
Colloquium Mathematicae
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