Geodesic
Hereditarily Indecomposable Hausdorff Continua Have Unique Hyperspaces $2^X$ and $C_n(X)$
Publications de l'Institut Mathématique, _N_S_89 (2011) no. 103, p. 49 .

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Let $X$ be a Hausdorff continuum (a compact connected Hausdorff space). Let $2^X$ (respectively, $C_n(X)$) denote the hyperspace of nonempty closed subsets of $X$ (respectively, nonempty closed subsets of $X$ with at most $n$ components), with the Vietoris topology. We prove that if $X$ is hereditarily indecomposable, $Y$ is a Hausdorff continuum and $2^X$ (respectively $C_n(X)$) is homeomorphic to $2^Y$ (respectively, $C_n(Y) $), then $X$ is homeomorphic to $Y$.
Classification : 54B20
Keywords: generalized arc, Hausdorff continuum, hereditarily indecomposable, hyperspace, unique hyperspace, Vietoris topology 
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     author = {Alejandro Illanes},
     title = {Hereditarily {Indecomposable} {Hausdorff} {Continua} {Have} {Unique} {Hyperspaces} $2^X$ and $C_n(X)$},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {49 },
     publisher = {mathdoc},
     volume = {_N_S_89},
     number = {103},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_2011_N_S_89_103_a4/}
}
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Alejandro Illanes. Hereditarily Indecomposable Hausdorff Continua Have Unique Hyperspaces $2^X$ and $C_n(X)$. Publications de l'Institut Mathématique, _N_S_89 (2011) no. 103, p. 49 . http://geodesic.mathdoc.fr/item/PIM_2011_N_S_89_103_a4/