Geodesic
Disconnectedness properties of hyperspaces
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 4, pp. 569-591 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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     author = {Hern\'andez-Guti\'errez, Rodrigo and Tamariz-Mascar\'ua, Angel},
     title = {Disconnectedness properties of hyperspaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {569--591},
     year = {2011},
     volume = {52},
     number = {4},
     mrnumber = {2864000},
     zbl = {1249.54024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2011_52_4_a8/}
}
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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/