Concerning exactly (𝑛,1) images of continua

Nadler, Sam B.; Ward, L. E.

doi:10.1090/s0002-9939-1983-0681847-3

Cited by 16 publications

(12 citation statements)

References 11 publications

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“…It is proved that no hereditarily indecomposable tree-like continuum is the continuous exactly k-to-1 image of any continuum if k ≥ 2. This result gives more information towards a resolution of a question posed by Nadler and Ward [10] i.e., which continua are k-to-1 images of continua, where k ≥ 2? The result also generalizes a result of J. Heath [5] who proved that no hereditarily indecomposable tree-like continuum is a two-to-one image of a continuum.…”

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confidence: 80%

Exactly 𝑘-to-1 maps and hereditarily indecomposable tree-like continua

Gonzalez

2003

Proc. Amer. Math. Soc.

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Abstract. In 1947, W.H. Gottschalk proved that no dendrite is the continuous, exactly k-to-1 image of any continuum if k ≥ 2. Since that time, no other class of continua has been shown to have this same property. It is shown that no hereditarily indecomposable tree-like continuum is the continuous, exactly k-to-1 image of any continuum if k ≥ 2.One of the earliest results concerning exactly k-to-1 maps between continua is W. H. Gottschalk's [2] result that no dendrite is the continuous exactly k-to-1 image of any continuum if k ≥ 2. Since Gottschalk's result, no other class of continua has been shown to repel exactly k-to-1 functions from continua in the manner that dendrites do. It is proved that no hereditarily indecomposable tree-like continuum is the continuous exactly k-to-1 image of any continuum if k ≥ 2. This result gives more information towards a resolution of a question posed by Nadler and Ward [10] i.e., which continua are k-to-1 images of continua, where k ≥ 2? The result also generalizes a result of J. Heath [5] who proved that no hereditarily indecomposable tree-like continuum is a two-to-one image of a continuum. It is known that for each k > 2 there exists a k-to-1 map between tree-like continua [4]. For more results concerning k-to-1 functions between continua, the reader is directed to a survey paper of J. Heath [7].A space is a compact metric space, a continuum is a nonempty, compact, connected metric space, and a map is a continuous function. If X and Y are spaces, then a map f from X into Y is said to be confluent if for any continuum L in the image, every component of f −1 (L) maps onto L. Lemma 1. Suppose that f is a confluent map onto a space

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confidence: 80%

Exactly 𝑘-to-1 maps and hereditarily indecomposable tree-like continua

Gonzalez

2003

Proc. Amer. Math. Soc.

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“…Some known theorems and the theorems in this section will explain the "probably" . For instance, Nadler and Ward [10] showed that if a continuum Y fails to be hereditarily unicoherent, then Y is a 2-to-1 retract (of a continuum). So if there is a simple closed curve in Y for instance, or a Warsaw circle, then Y is a 2-to-1 retract.…”

Section: Non-treelike Continua That Are 2-to-1 Retracts Of Continuamentioning

confidence: 99%

A non-treelike continuum that is not the 2-to-1 image of any continuum

Heath

1996

Proc. Amer. Math. Soc.

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Abstract. Some thirteen years ago S. B. Nadler, Jr. and L. E. Ward, Jr., asked if any treelike continuum could be the 2-to-1 image of a continuum. In fact, it has been conjectured that the property of being treelike characterizes those continua that are not the 2-to-1 image of any continuum. But the characterization must be something else; this paper shows that many pseudosolenoids are not the 2-to-1 image of any continuum.

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“…In 1983 Sam Nadler and Lew Ward [8] asked if a tree-like continuum can be the image of a 2-to-1 map (defined on a continuum, of course). That question is answered only for some cases.…”

Section: Introductionmentioning

confidence: 99%