Centers of a dendroid

Abstract

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.1. Introduction. The point p is a center of the dendroid D if there are two points c and d in D such that for every ε > 0 there is a continuum C containing p of diameter less than ε and there are two open sets, U containing c and V containing d, such that C intersects every arc from U to V . The concept of a dendroid center was introduced by Piotr Minc and he proved that every dendroid contains at least one center [5, Theorem 3.6]. We used the results of Minc in [1] to show that no continuum maps exactly 2-to-1 onto a dendroid. Minc called the continuum C a bottleneck, the points c and d basin points, and the two open sets U and V basins.We can put all dendroids into distinct and useful classes with strong structural elements determined by the behavior of the centers or the only center. We show that the set of centers of a dendroid is arc connected and a dendroid with more than one center has uncountably many strong centers. We find that every non-endpoint is a strong center in a locally connected dendroid (also called a dendrite), and that the strong centers are dense in a Suslinean dendroid. On the other hand, some dendroids have only one center. For example, the cone point in the cone over a Cantor set is the only center and it is a strong center. A significantly more complicated dendroid