Geodesic
Disconnectedness properties of hyperspaces
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 4, pp. 569-591.

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Let $X$ be a Hausdorff space and let $\mathcal H$ be one of the hyperspaces $CL(X)$, $\mathcal K(X)$, $\mathcal F(X)$ or $\mathcal F_n(X)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathcal H$: extremal disconnectedness, being a $F'$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that (a) both $F$ and $X-F$ are totally disconnected, (b) the quotient $X/F$ is hereditarily disconnected, then $\mathcal K(X)$ is hereditarily disconnected. We also show an example proving that this result cannot be reversed.
Classification : 54B20, 54G05, 54G10, 54G12, 54G20
Keywords: hyperspaces; Vietoris topology; $F'$-space; $P$-space; hereditarily disconnected 
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Hernández-Gutiérrez, Rodrigo; Tamariz-Mascarúa, Angel. Disconnectedness properties of hyperspaces. Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 4, pp. 569-591. http://geodesic.mathdoc.fr/item/CMUC_2011__52_4_a8/