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 $n\geq 4.$ Our result generalize all the previous known results on this subject.

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