Disconnectedness properties of hyperspaces

Rodrigo Hernández-Gutiérrez; Angel Tamariz-Mascarúa

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 4, page 569-591
  • ISSN: 0010-2628

Abstract

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Let X be a Hausdorff space and let ℋ be one of the hyperspaces C L ( X ) , 𝒦 ( X ) , ℱ ( X ) or ℱ n ( X ) ( n a positive integer) with the Vietoris topology. We study the following disconnectedness properties for 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 ⊂ 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 show an example proving that this result cannot be reversed.

How to cite

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Hernández-Gutiérrez, Rodrigo, and Tamariz-Mascarúa, Angel. "Disconnectedness properties of hyperspaces." Commentationes Mathematicae Universitatis Carolinae 52.4 (2011): 569-591. <http://eudml.org/doc/246289>.

@article{Hernández2011,
abstract = {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 $F^\{\prime \}$-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.},
author = {Hernández-Gutiérrez, Rodrigo, Tamariz-Mascarúa, Angel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {hyperspaces; Vietoris topology; $F^\{\prime \}$-space; $P$-space; hereditarily disconnected; hyperspace; Vietoris topology; -space; -space; hereditary disconnectedness},
language = {eng},
number = {4},
pages = {569-591},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Disconnectedness properties of hyperspaces},
url = {http://eudml.org/doc/246289},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Hernández-Gutiérrez, Rodrigo
AU - Tamariz-Mascarúa, Angel
TI - Disconnectedness properties of hyperspaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 4
SP - 569
EP - 591
AB - 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 $F^{\prime }$-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.
LA - eng
KW - hyperspaces; Vietoris topology; $F^{\prime }$-space; $P$-space; hereditarily disconnected; hyperspace; Vietoris topology; -space; -space; hereditary disconnectedness
UR - http://eudml.org/doc/246289
ER -

References

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