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Displaying similar documents to “Continua with countable number of arc-components”

No arc-connected treelike continuum is the 2-to-1 image of a continuum

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.

Centers of a dendroid

Jo Heath, Van C. Nall (2006)

Fundamenta Mathematicae

Similarity:

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,...

Exactly two-to-one maps from continua onto some tree-like continua

Wojciech Dębski, J. Heath, J. Mioduszewski (1992)

Fundamenta Mathematicae

Similarity:

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...