Hereditarily Indecomposable Hausdorff Continua Have Unique Hyperspaces $2^x$ and $c_n(X)$

Alejandro Illanes

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 $Y$.