Rigidity of symmetric products

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].…”

“…It has been proved by Hernández-Gutiérrez and Martínez-de-la-Vega [11] that wired continua have unique hyperspace F n (X), for n ≥ 4. The authors of this paper prove that meshed continua have unique hyperspace F n (X), for each n ∈ {2, 3}.…”

Homogeneity degree of some symmetric products

“…Given a continuum X and n ≥ 2, the hyperspace F n (X) is rigid provided that for each h ∈ H(F n (X)), h(F 1 (X)) = F 1 (X). Rigidity of symmetric products was studied in [3], where it was shown that ([3, Theorem 17]) if X is an m-manifold, m ≥ 2 and n ≥ 3, then F n (X) is rigid. Using a theorem by R. Molski [8], we show that this result is also true for m ≥ 3 and n = 2.…”

“…By Theorem 17 of [3], if n ≥ 3, then F n (X) is rigid. So, if h ∈ H(F n (X)), then h(F 1 (X)) = F 1 (X).…”

“…In the case that m ≥ 2 and n ≥ 3, Theorem 17 in [3] 1 . Moreover, by the first part of the proof of Theorem 17 in [3], we have…”