Uniqueness of hyperspaces for Peano continua

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].…”

“…We recall that, as in [9], a continuum X is said to be almost meshed provided that the set G(X) is dense in X; and an almost meshed continuum X is meshed provided that X has a basis of neighborhoods B such that U − P(X) is connected, for each element U ∈ B. Let AM = {X: X is an almost meshed continuum}, and let M = {X: X is a meshed continuum}.…”

“…A locally D-continuum is a continuum such that each point has a basis of neighborhoods B such that each element in B is an element of D. Let LD = {X: X is a locally D-continuum}. Notice that LD M (see [9,Theorem 6]). In the Fifth Research Workshop in Hyperspaces and Continuum Theory, that took place in the Mexico City, 2011, Alejandro Illanes asked if X is a locally D-continuum, does X have unique hyperspace F n (X), for each n ∈ {2, 3}?…”

Hereditarily indecomposable Hausdorff continua have unique hyperspaces 2X and Cn(X)

“…Uniqueness of hyperspaces has been widely studied (see [2], [3, p. 285] and [4]). Macías proved in [7] and [8,Theorem 6.1] that if X is a metric hereditarily indecomposable Hausdorff continuum, then X has unique hyperspaces 2 X and C n (X) (for each n).…”