Geodesic
Continua with unique symmetric product
Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 3, pp. 397-406.

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Let $X$ be a metric continuum. Let $F_{n}(X)$ denote the hyperspace of nonempty subsets of $X$ with at most $n$ elements. We say that the continuum $X$ has unique hyperspace $F_{n}(X)$ provided that the following implication holds: 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 prove the following results: (1) if $X$ is an indecomposable continuum such that each nondegenerate proper subcontinuum of $X$ is an arc, then $X$ has unique hyperspace $F_{2}(X)$, and (2) let $X$ be an arcwise connected continuum for which there exists a unique point $v\in X$ such that $v$ is the vertex of a simple triod. Then $X$ has unique hyperspace $F_{2}(X)$.
Classification : 54B20, 54F15
Keywords: arc continuum; continuum; indecomposable; symmetric product; unique hyperspace 
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Anaya, José G.; Castañeda-Alvarado, Enrique; Illanes, Alejandro. Continua with unique symmetric product. Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 3, pp. 397-406. http://geodesic.mathdoc.fr/item/CMUC_2013__54_3_a6/