Abstract
We continue in the study of blockers in continua that were first defined by Illanes and Krupski. Especially, we are dealing with the following question of these authors.
For a given continuum, if each closed set that blocks any finite set also blocks any closed set, does it imply that the continuum is locally connected? We provide a negative answer by constructing a planar non-locally connected lambda-dendroid in which every closed set which blocks every finite set also blocks every closed set. On the other hand we prove that in the realm of hereditarily decomposable chainable continua or among smooth dendroids the answer is positive.
Finally we compare the notion of a non-blocker with the notion of a shore set and we show that the union of finitely many mutually disjoint closed shore sets in a smooth dendroid is a shore set. This answers a question of the authors.