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Centers of a dendroid

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Heath J., Nall V.

Fundamenta Mathematicae

2006

tom 189

nr 2

173-183

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, every open set intersects uncountably many of these arc components. Moreover, we find that a map from one dendroid to another preserves the center structure if each point inverse has at most countably many components.

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

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Heath J., Nall V.

Fundamenta Mathematicae

2003

tom 180

nr 1

11-24

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.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Heath J.

2 Nall V.

1 2006

1 2003

Wyniki wyszukiwania

Centers of a dendroid

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