Making holes in hyperspaces

Abstract

Let X be a metric continuum and C(X) the hyperspace of all nonempty subcontinua of X. Let A ∈ C(X), A is said to make a hole in C(X), if C(X) − {A} is not unicoherent. In this paper we study the following problem. Problem: For which A ∈ C(X), A makes a hole in C(X).In this paper we present some partial solutions to this problem in the following cases: (1) A is a free arc; (2) A is a one-point set;(3) A is a free simple closed curve; (4) A = X.

Making holes in the cone, suspension and hyperspaces of some continua

“…So it becomes interesting to determine the elements that make a hole in these hyperspaces. In this direction, see [1], [2] and [4], for what is known about this problem. This section is dedicated to giving an application in hyperspaces of what we have developed.…”

“…Let Z be a metric space. The following are normal spaces:a) Z × I, b) Cone(Z), c) Sus(Z), d) Cone(Z) − {[z, t] c } for z ∈ Z and t ∈ [0,1), and e) Sus(Z) − {[z, t] s } for z ∈ Z and t ∈ (0, 1).Proof: For a), since Z is a metric space, so is Z × I. Thus Z × I is a normal space.…”

“…))}; 5) the double Warsaw circle S1 3 = S 0 ∪{(x, sin(1/x)) :x ∈ [−1, 0)}∪A, where A is an arc from (−1, sin(−1)) to (1, sin(1)), such thatS 0 ∪ x, sin 1 x : x ∈ [−1, 0) ∩ A = {(−1, sin(−1)), (1, sin(1))}; 6) (SP ) 1 , see Example 2.14; 7) (SP ) 2 = (SP ) 1 ∪ {(1 − 1/t) exp(it) : t ≥ 1} with the points 2 exp(i) and (0, 0) identified; 8) (SP ) 3 = (SP ) 1 ∪ {(1 + 1/t) exp(it) : t ≤ −1} with the points 2 exp(i) and…”

Making Holes in the Second Symmetric Product of a Cyclicly Connected Graph

“…876, 877) and (Michael, 1951, Theorem 4.10, p. 165 (Y). This is a generalization of the notion of to make a hole in a unicoherent topological space defined in (Anaya, 2007(Anaya, , p. 2000.…”

“…Readers specially interested in this problem are refered to Anaya (2007Anaya ( , 2011, Orozco-Zitli (2010, 2012).…”