Peano Continua with Unique Symmetric Products

Abstract

Let $X$ be a metric continuum and $n$ a positive integer. Let $F_{n}(X)$ be the hyperspace of all nonempty subsets of $X$ with at most $n$ points, metrized by the Hausdorff metric. We said that $X$ has unique hyperspace $F_n(X)$ provided that, if $Y$ is a continuum and $F_n(X)$ is homeomorphic to $F_n(Y),$ then $X$ is homeomorphic to $Y.$ In this paper we study Peano continua $X$ that have unique hyperspace $F_n(X)$, for each $ngeq 4.$ Our result generalize all the previous known results on this subject

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].…”

“…(b) If X ∈ D and n ∈ N, then X has unique hyperspace F n (X) (see [2,Theorem 5.2], [15,Theorem 3.7]). …”

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