Blockers in hyperspaces

The hyperspace of nonblockers of F1(X)

“…Note that if Λ(q, B) ∩ A n = ∅, then there is K ∈ C(X) such that q ∈ K, K ∪ A n ∈ C(X) and {x, y} ∩ (K ∪ A n ) = ∅. Hence, {x, y} does not block {q}, by Lemma 5.1 (6). A contradiction.…”

“…Hence, using Lemma 5.1, it is not difficult to see that there exists a subcontinuum L of X such that {q} ∪ A n ⊆ L, and L ∩ {x, y} = ∅. Thus, {x, y} does not block {q}, by Lemma 5.1 (6). A contradiction.…”

“…Thus, α(t 0 ) ∪ W is a subcontinuum of X such that int X (α(t 0 ) ∪ W ) = ∅ and {x, y} ∩ (α(t 0 ) ∪ W ) = ∅. Therefore, {x, y} does not block {z}, by Lemma 5.1 (6). This contradicts that N B(F 1 (X)) = F 1 (X) and int X (K) = ∅ for each indecomposable subcontinuum K of X.…”

“…The notions of blocker and nonblocker in hyperspaces have been studied recently by many authors (see [2], [3], [4], [5] and [6]). In [6], the authors introduce the notion of blocker, and present general properties using different kind of continua. In [4], the set N B(F 1 (X)) is used to characterizes classes of continua; for instance, it is proved that if X is localy connected, then N B(F 1 (X)) = F 1 (X) if and only if X is the simple closed curve.…”

Non-cut, shore and non-block points in continua

“…More authors have investigated various properties of special sets in continua: Grace in [Gr81] provides a survey of results relating the notions of aposyndesis and weak cut point; Illanes in [Il01] shows that, in a dendroid, finite union of pairwise disjoint shore subdendroids is a shore set; among other results, a simple example of a planar dendroid in which the union of two disjoint closed shore sets is not a shore set is presented in [BMPV14]; in [Na07] Nall explores the relationship between center points and shore points in a dendroid; Illanes and Krupski study blockers and nonblockers for several kinds of continua ( [IKr11]); and, using the results of [IKr11], Escobedo, López and Villanueva ( [ELV12]) characterize some classes of locally connected continua -for further information on the subject see also [PV12,Le13].…”