Bottlenecks in dendroids

A subcontinuum C of a dendroid X is a bottleneck if it intersects every arc connecting two nonempty open subsets of X. We prove that every dendroid has a point p contained in arbitrarily small bottlenecks. Moreover, every plane dendroid contains a single point bottleneck. This implies, for instance, that each map from an indecomposable continuum into a plane dendroid must have an uncountable point-inverse. We also construct a map from the simplest indecomposable continuum into a (nonplanar) dendroid with all point-inverses consisting of at most three points.