Local dendrites with unique n-fold hyperspace

Abstract

Let Z be a metric continuum and n be a positive integer. Let C n (Z ) be the hyperspace of the nonempty closed subsets of Z with at most n components. In this paper we prove the following result: Let X be a local dendrite such that every point of X has a neighborhood which is a dendrite whose set of end points is closed and Z is any continuum such that C n (X) is homeomorphic to C m (Z ) for some n, m 3, then X is homeomorphic to Z .

Meshed continua have unique second and third symmetric products

“…We use, adapt and generalize results that have been published in the area of uniqueness of hyperspaces, the more related ones can be found in [1][2][3][4]6,7] and [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

“…Guerrero-Méndez et al / Topology and itsApplications 191 (2015) [16][17][18][19][20][21][22][23][24][25][26][27] …”

Peano Continua with Unique Symmetric Products

“…4, No. 4;2012 Assume that x ∈ X − (P(X) ∪ R(X)). Then there exists a finite graph T in X such that x ∈ int X (T ).…”

Dendrites with unique hyperspace C2(X)

“…E. Castañeda and A. Illanes [9] have proved that finite graphs have unique hyperspace F n (X) for each n. A. Illanes proved that ( [18] and [19]) finite graphs have unique hyperspaces C n (X), for each n 1. Related results to the subject of this paper can be found in [1][2][3][4][5][6][12][13][14][15][16][17][18]21] and [22]. The non-metric case has been considered in [24].…”

“…In [13] and [14] it has been shown that if X is a dendrite with closed set of end points, then X has unique hyperspace C n (X) for each n = 2. This case can be proved by using the fact that it is possible to give topological properties that characterize the set F 1 (X) inside C n (X).…”