Centers of a dendroid

Jo Heath¹ ; Van C. Nall²

¹ Mathematics Department Parker Hall Auburn University Auburn, AL 36849-5310, U.S.A.
² Department of Mathematics and Computer Science University of Richmond Richmond, VA 23173, U.S.A.

Fundamenta Mathematicae, Tome 189 (2006) no. 2, pp. 173-183

Résumé

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.

Détail

DOI : 10.4064/fm189-2-6

Mots-clés : bottleneck dendroid continuum intersects every arc connecting non empty sets piotr minc proved every dendroid contains point which call center contained arbitrarily small bottlenecks study effect set centers dendroid has its structure set centers arc connected dendroid only center has uncountably many arc components complement center every set intersects uncountably many these arc components moreover map dendroid another preserves center structure each point inverse has countably many components