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Disconnectedness properties of hyperspaces

Rodrigo Hernández-Gutiérrez, Angel Tamariz-Mascarúa (2011)

Commentationes Mathematicae Universitatis Carolinae

Let X be a Hausdorff space and let be one of the hyperspaces C L ( X ) , 𝒦 ( X ) , ( X ) or n ( X ) ( n a positive integer) with the Vietoris topology. We study the following disconnectedness properties for : 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 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 𝒦 ( X ) is hereditarily disconnected. We also...

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