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hyperspaces; Vietoris topology; $F'$-space; $P$-space; hereditarily disconnected
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.
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