Centers of a dendroid
Volume 189 / 2006
Fundamenta Mathematicae 189 (2006), 173-183
MSC: Primary 54F50; Secondary 54F15.
DOI: 10.4064/fm189-2-6
Abstract
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.
