On the hyperspace of subcontinua of a finite graph I

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].…”

The hyperspaces Cn(X) for finite ray-graphs

“…For our base spaces, we will consider a variation on finite graphs, which we will call finite raygraphs. These graphs will consist of a finite number of vertices (points), edges (homeomorphic to [0,1] and attached at two vertices, or at one vertex twice) and rays (homeomorphic to [0, ∞) and attached at one vertex). We will restrict our attention to finite connected ray-graphs.…”

“…In 1968, Duda did an examination of the hyperspace of subcontinua of finite connected graphs, and under some minor conditions was able to give a description of the hyperspace as a polyhedron, decomposable into balls of various dimensions [1], [2]. A single hyperspace may consist of several sections of different dimension: a two-dimensional disc glued to a three dimensional ball, etc.…”

Dendrites with unique hyperspace C2(X)

“…(a) Finite graphs different from an arc and a simple closed curve [10, 9.1] and [11]. (b) Hereditarily indecomposable continua [29, 0.60].…”