Nonblockers in hyperspaces

The hyperspace of nonblockers of F1(X)

“…It is not difficult to see that s(x) = {L n : n ∈ N}. Thus, we prove (4). Finally, notice that s(x) is a F σ -set, for each x ∈ X \ {p}.…”

“…Theorem 5.3 shows that the simple closed curve is the unique continuum X such that the hyperspace of nonblockers of F 1 (X) is in fact, F 1 (X). Thus, we generalize [4,Theorem 3.2], give a positive answer to Question 1.1, complete Theorem 1.2 and answer Question 1.3 negatively.…”

“…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. Naturally, they posed the following question: Question 1.1.…”

“…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.…”

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].…”