On reflexive closed set lattices

Zhongqiang Yang; Dong Sheng Zhao

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 1, page 143-154
  • ISSN: 0010-2628

Abstract

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For a topological space X , let S ( X ) denote the set of all closed subsets in X , and let C ( X ) denote the set of all continuous maps f : X → X . A family 𝒜 ⊆ S ( X ) is called reflexive if there exists 𝒞 ⊆ C ( X ) such that 𝒜 = { A ∈ S ( X ) : f ( A ) ⊆ A for every f ∈ 𝒞 } . Every reflexive family of closed sets in space X forms a sub complete lattice of the lattice of all closed sets in X . In this paper, we continue to study the reflexive families of closed sets in various types of topological spaces. More necessary and sufficient conditions for certain families of closed sets to be reflexive are obtained.

How to cite

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Yang, Zhongqiang, and Zhao, Dong Sheng. "On reflexive closed set lattices." Commentationes Mathematicae Universitatis Carolinae 51.1 (2010): 143-154. <http://eudml.org/doc/37739>.

@article{Yang2010,
abstract = {For a topological space $X$, let $S(X)$ denote the set of all closed subsets in $X$, and let $C(X)$ denote the set of all continuous maps $f:X\rightarrow X$. A family $\mathcal \{A\}\subseteq S(X)$ is called reflexive if there exists $\{\mathcal \{C\}\}\subseteq C(X)$ such that $\mathcal \{A\} = \lbrace A\in S(X) : f(A)\subseteq A$ for every $f\in \{\mathcal \{C\}\}\rbrace $. Every reflexive family of closed sets in space $X$ forms a sub complete lattice of the lattice of all closed sets in $X$. In this paper, we continue to study the reflexive families of closed sets in various types of topological spaces. More necessary and sufficient conditions for certain families of closed sets to be reflexive are obtained.},
author = {Yang, Zhongqiang, Zhao, Dong Sheng},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {reflexive families of closed sets; closed set lattice; hyperspace; lower semicontinuous set-valued map; reflexive family; closed set lattice; hyperspace; lower semicontinuous set-valued map},
language = {eng},
number = {1},
pages = {143-154},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On reflexive closed set lattices},
url = {http://eudml.org/doc/37739},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Yang, Zhongqiang
AU - Zhao, Dong Sheng
TI - On reflexive closed set lattices
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 1
SP - 143
EP - 154
AB - For a topological space $X$, let $S(X)$ denote the set of all closed subsets in $X$, and let $C(X)$ denote the set of all continuous maps $f:X\rightarrow X$. A family $\mathcal {A}\subseteq S(X)$ is called reflexive if there exists ${\mathcal {C}}\subseteq C(X)$ such that $\mathcal {A} = \lbrace A\in S(X) : f(A)\subseteq A$ for every $f\in {\mathcal {C}}\rbrace $. Every reflexive family of closed sets in space $X$ forms a sub complete lattice of the lattice of all closed sets in $X$. In this paper, we continue to study the reflexive families of closed sets in various types of topological spaces. More necessary and sufficient conditions for certain families of closed sets to be reflexive are obtained.
LA - eng
KW - reflexive families of closed sets; closed set lattice; hyperspace; lower semicontinuous set-valued map; reflexive family; closed set lattice; hyperspace; lower semicontinuous set-valued map
UR - http://eudml.org/doc/37739
ER -

References

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